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Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity
International Journal of Solids and Structures 40 (2003) 251–269 www.elsevier.com/locate/ijsolstr
Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity Chein-Shan Liu
*
Department of Mechanical and Marine Engineering, National Taiwan Ocean University, Keelung 202-24, Taiwan Received 24 May 2002; received in revised form 25 September 2002
Abstract The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which Xa0 and Xc0 play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector Xa :¼ ðY Qta ; YQ0a Þt is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime Mnþ1 on which the proper orthochronous Lorentz group SOo ðn; 1Þ left acts. Then, constructed is a Poincare group ISOo ðn; 1Þ on space X :¼ Xa þ Xb , of which Xb reflects the kinematic hardening rule in the model. We also find that the space ðQta ; qa0 Þ is a Robertson–Walker spacetime, which is conformal to Xa through a factor Y , and conformal to Xc :¼ ðqQta ; qQ0a Þt through a factor q as given by qðqa0 Þ ¼ Y ðqa0 Þ=½1 2q0 Q0a ð0Þ þ 2q0 Y ðqa0 ÞQ0a ðqa0 Þ. In the conformal spacetime the internal symmetry is a conformal group. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elastoplasticity; Mixed-hardening; Symmetry group; Minkowski spacetime; Conformal spacetime
1. Introduction The formulation of yield criteria of Tresca and of von Mises was the first key step in developing plasticity theory. However, without a plastic stress-strain relation, the criteria can do little. The incremental stressstrain relations proposed by Saint-Venant and Levy represented a giant step forward in plasticity theory. Since then the incremental or rate type point-of-view has been most frequently taken in formulating constitutive equations of plasticity, which describe the evolutions of internal state variables. This kind of rate-equation representations is now composed of separately specified but interwoven ingredients, including the elastic part, yield condition, loading-unloading criterion, plastic flow rule, isotropic hardening rule and kinematic hardening rule, etc. Because of the nonlinear nature of the model ingredients in plasticity, there seems necessary although difficult by composing all ingredients together to derive a global theory of plasticity, which not only gives
*
Tel.: +886-2-2462-2192x3252; fax: +886-2-2462-0836. E-mail address: [email protected] (C.-S. Liu).
0020-7683/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 2 ) 0 0 5 5 2 - 8
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directly a local response of the material in the time domain once an input is prescribed, but also facilitates us to handle model property from a global view. To do this several thresholds need to be stridden across. The first is integrating the rate-type equations, which consist of separated but interwoven ingredients specified on a high-dimensional space such as states in stress space and paths in plastic strain space and their product space. The second is solving the irreversibility parameters, such as the equivalent plastic strain, dissipation, and so on, and in general there exist monotonic mappings between these parameters. The third as to be strongly emphasized here is studying symmetry properties of the model and its intrinsic spacetime structures. The differential nature of plasticity laws has been discussed for a long time. The invariant yield condition in stress space renders a natural mathematical frame of plasticity theory from the view of differentiable manifold and its Lie group transformation. In this paper we are going to make an effort to study the internal symmetry groups and the underlying spacetime structures for mixed-hardening elastoplasticity. The Lie group theory provides a universal tool for tackling considerable numbers of differential equations when other means fail. Indeed, group analyses may augment intuition in understanding and in using symmetry for formulation of physical models, and often disclose possible approaches to solving complex problems. Recently, Hong and Liu (1999a,b, 2000) have considered constitutive models of perfect elastoplasticity with and without considering large deformation and of bilinear elastoplasticity, revealing that the models possess two kinds of internal symmetries, characterized by the translation group T ðnÞ in the off (or elastic) phase and by the projective proper orthochronous Lorentz group PSOo ðn; 1Þ in the on (or elastoplastic) phase for the first two models and the proper orthochronous Poincare group PISOo ðn; 1Þ in the on phase for the bilinear model, and have symmetry switching between the two groups dictated by the control input. Moreover, Liu (2001, 2002) applied a PDSOo ðn; 1Þ symmetry to formulate a mathematical model of visco-elastoplasticity, and then Liu (submitted for publication) employed a two-component spinor representation to unify finite strain elastic-perfectly plastic models with different objective stress rates and showed that the Lie symmetry in the two-dimensional spinor space is SLð2; HÞ with H denoting the quaternionic number. In this paper some of those results are extended to the model of mixed-hardening elastoplasticity, investigating the influence of nonlinear hardening parameters on the structure of the underlying vector spaces and on the transformation groups of internal symmetries.
2. Constitutive postulations In terms of five-dimensional stress vectors and five-dimensional strain vectors: 3 3 3 2 11 2 11 2 11 a1 sa þ a2 s22 a1 sb þ a2 s22 a1 s þ a2 s22 a b 6 a3 s11 þ a4 s22 7 6 a3 s11 þ a4 s22 7 6 a3 s11 þ a4 s22 7 7 6 6 a 23 a 7 6 b 23 b 7 23 7 7 7; 6 6 ; Q ; Q Q¼6 ¼ ¼ s sb s a b a 7 7 7 6 6 6 13 13 13 5 5 5 4 4 4 s s s a b 12 s12 s s12 a b 2
3 a1 e11 þ a2 e22 6 a3 e11 þ a4 e22 7 6 7 7; q¼6 e23 6 7 4 5 e13 e12
2
3 a1 ee11 þ a2 ee22 6 a3 ee11 þ a4 ee22 7 6 7 e 7; ee23 q ¼6 6 7 4 5 ee13 ee12
3 a1 ep11 þ a2 ep22 6 a3 ep11 þ a4 ep22 7 6 7 p 7; q ¼6 ep23 6 7 4 5 ep13 ep12
ð1Þ
2
ð2Þ
the tensorial form of elastoplastic model with mixed-hardening rule, e.g., Hong and Liu (1993): e_ ¼ e_ e þ e_ p ;
ð3Þ
C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
253
s ¼ sa þ sb ;
ð4Þ
s_ ¼ 2G_ee ;
ð5Þ
k_sa ¼ 2s0y e_ p ;
ð6Þ
s_ b ¼ 2k 0 e_ p ;
ð7Þ
pffiffiffi ksa k 6 2s0y ;
ð8Þ
k_ P 0;
ð9Þ
pffiffiffi ksa kk_ ¼ 2s0y k_;
ð10Þ
can be equivalently re-expressed by the following vectorial form: q_ ¼ q_ e þ q_ p ;
ð11Þ
Q ¼ Qa þ Qb ;
ð12Þ
_ ¼ ke q_ e ; Q
ð13Þ
Qa q_ a0 ¼ Q0a q_ p ;
ð14Þ
_ b ¼ kp q_ p ; Q
ð15Þ
kQa k 6 Q0a ;
ð16Þ
q_ a0 P 0;
ð17Þ
kQa kq_ a0 ¼ Q0a q_ a0 :
ð18Þ
In Eqs. (1) and (2), p a1 :¼ sin h þ ; 3
p a3 :¼ cos h þ ; 3
ð19Þ pffiffiffi 11 where h can be any real number. If choosing h ¼ 0 we have the stress space Q :¼ ð 3s =2; s11 =2 þ s22 ; t s23 ; s13 ; s12 Þ of IlÕyushin (1963). Throughout this paper, a superscript t indicates the transpose, and a superimposed dot denotes a differentiation with respect to time. In the tensorial representation of elastoplastic model with PragerÕs kinematic hardening rule combined with isotropic hardening rule as shown by Eqs. (3)–(10), e, ee , ep , s, sa and sb are, respectively, the deviatoric tensors of strain, elastic strain, plastic strain, stress, active stress and back stress, all symmetric and traceless, whereas k is a scalar evaluated by Z t pffiffiffi kðtÞ ¼ 2k_ep ðnÞk dn; ð20Þ a2 :¼ sin h;
a4 :¼ cos h;
0
pffiffiffiffiffiffiffiffiffiffiffiffi where k_e k :¼ e_ p e_ p defines the Euclidean norm of e_ p and a dot between two tensors stands for their inner product. From Eqs. (1) and (2) it follows that p
2
2
ksa k ¼ 2kQa k ;
sa e_ ¼ 2Qta q_ :
ð21Þ
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The material functions and plastic multipliers in the above two equivalent representations have the following relations: ke ¼ 2G;
kp ¼ 2k 0 ;
Q0a ¼ s0y ;
2q_ a0 ¼ k_:
ð22Þ ffi pffiffiffiffiffiffiffiffiffi The norm of an n-vector Q is defined as kQk :¼ Qt Q. Depending on the number of nonzero stress components in Eq. (1) (and correspondingly nonzero strain components in Eq. (2)) which we consider for a physical problem, for example, the axial tension-compression problem, the biaxial tension-compressiontorsion problem, etc., the dimension n may be an integer with 1 6 n 6 5, and no matter which case is we use n to denote the physical problem dimension. The bold-faced letters q, qe , qp , Q, Qa and Qb are, respectively, the n-vectors of generalized strain, generalized elastic strain, generalized plastic strain, generalized stress, generalized active stress and generalized back stress, whereas qa0 is a scalar. All q, qe , qp , Qa , Qb , Q and qa0 are functions of one and the same independent variable, which in most cases is taken either as the usual time or as the arc length of the controlled generalized strain path; however, for convenience, the independent variable no matter what it is will be simply called ‘‘time’’ and given the notation t. The generalized elastic modulus ke > 0, generalized kinematic modulus kp , and the generalized yield strength Q0a > 0 are the only three property functions needed in the model, and are functions of the equivalent generalized plastic strain qa0 , which by Eqs. (14), (17) and (18) and Q0a > 0 is found to be Z t Z t kq_ p ðnÞk dn: ð23Þ q_ a0 ðnÞ dn ¼ qa0 ¼ 0
0
The material is further assumed to be weakly stable (Hong and Liu, 1993): 0
ke þ kp þ Q0a > 0;
ke þ kp > 0:
ð24Þ
Here a prime denotes the derivative of a function with respect to its argument, for example, 0 Q0a :¼ dQ0a ðqa0 Þ=dqa0 .
3. Response operators From Eqs. (11)–(15) it follows that _ a þ ke þ kp q_ a Qa ¼ ke q_ ; Q 0 Q0a which, in terms of the integrating factor Z qa
0 k ðpÞ þ k ðpÞ e p a Y ðq0 Þ :¼ exp dp ; Q0a ðpÞ 0
ð25Þ
ð26Þ
can be integrated to Qa ðtÞ ¼
Z t 1 a a a _ Y ðq ðt ÞÞQ ðt Þ þ k ðq ðnÞÞY ðq ðnÞÞ q ðnÞ dn : e 0 a i 0 i 0 Y ðqa0 ðtÞÞ ti
Upon substituting Eq. (27) for Qa into Eq. (14) and integrating the resultant, we obtain Z t Gp ðqa0 ðtÞ; qa0 ðti ÞÞ qp ðtÞ ¼ qp ðti Þ þ Q ðt Þ þ Gp ðqa0 ðtÞ; qa0 ðnÞÞq_ ðnÞ dn; a i ke ðqa0 ðti ÞÞ ti
ð27Þ
ð28Þ
C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
255
where Cðqa0 Þ :¼
Z
qa0
0
1 dp; Y ðpÞQ0a ðpÞ
Gp ðp1 ; p2 Þ :¼ ke ðp2 ÞY ðp2 Þ½Cðp1 Þ Cðp2 Þ:
ð29Þ ð30Þ
Integrating Eq. (13) after replaced its q_ e by q_ q_ p due to Eq. (11), q_ p by the flow rule (14) and Qa by Eq. (27), we obtain Z t Qa ðti Þ þ QðtÞ ¼ Qðti Þ þ Gs ðqa0 ðtÞ; qa0 ðti ÞÞ Gs ðqa0 ðti Þ; qa0 ðti ÞÞ Gs ðqa0 ðtÞ; qa0 ðnÞÞq_ ðnÞ dn; ð31Þ ke ðqa0 ðti ÞÞ ti where Ce ðqa0 Þ
Z
qa0
:¼ 0
ke ðpÞ dp; Y ðpÞQ0a ðpÞ
Gs ðp1 ; p2 Þ :¼ ke ðp2 Þf1 Y ðp2 Þ½Ce ðp1 Þ Ce ðp2 Þg:
ð32Þ ð33Þ
Up to now three integral operators (27), (28) and (31) are established which map respectively generalized strain rate histories into generalized active stress histories, generalized plastic strain histories and generalized stress histories. Here t is the (current) time and ti is an initial time, at which initial conditions Qa ðti Þ, qp ðti Þ, Qðti Þ and qa0 ðti Þ should be prescribed. Obviously, the expressions in Eqs. (27), (28) and (31) that the responses are in terms of the generalized strain rate history is useful only if the history of qa0 is already known. Thus qa0 deserves a more study, as will be pursued in the following two sections.
4. Switch of plastic irreversibility The complementary trios (16)–(18) enable the model to possess a switch of plastic irreversibility, the criteria for which are derived right below. Taking the inner product of Eq. (25) with Qa , we get bQta q_ ¼ Q0a q_ a0
if kQa k ¼ Q0a ;
ð34Þ
where b :¼
ke >0 ke þ kp þ Q0a0
ð35Þ
is a material function. Since b > 0 and Q0a > 0, Eq. (34) assures that if kQa k ¼ Q0a
then
Qta q_ > 0 () q_ a0 > 0:
ð36Þ
Hence, fkQa k ¼ Q0a and Qta q_ > 0g ) q_ a0 > 0:
ð37Þ
On the other hand, if q_ a0 > 0, Eq. (18) assures kQa k ¼ Q0a , which together with Eq. (36) asserts that q_ a0 > 0 ) fkQa k ¼ Q0a and Qta q_ > 0g:
ð38Þ
Therefore, from Eqs. (37) and (38) we conclude that the yield condition kQa k ¼ Q0a and the straining condition Qta q_ > 0 are sufficient and necessary for plastic irreversibility q_ a0 > 0.
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Considering this and the inequality (17), we thus reveal the following criteria of plastic irreversibility: ( t b _ > 0 if kQa k ¼ Q0a and Qta q_ > 0; 0 Q q a q_ 0 ¼ Qa a ð39Þ 0 if kQa k < Q0a or Qta q_ 6 0: In the phase of the switch, q_ a0 > 0, the mechanism of plastic irreversibility is working and the material exhibits elastoplastic behavior, while in the phase of the switch, q_ a0 ¼ 0, the material behavior is reversible and elastic. According to the complementary trios (16)–(18), there are just two phases: q_ a0 > 0 0 a 0 and kQa k ¼ Qa , and corresponds to the q_ 0 ¼ 0 and kQa k 6 Qa . From the switch (39) it is clear that phase whereas to the phase. 5. Measure of plastic irreversibility The switch (39) together with the relation (26) reveals the core importance of qa0 and Y in the whole model; we now go further to investigate their evolutions. Substitution of Eq. (27) into Eq. (34) and rearrangement yield Z t 1 0 a t a YQ q_ ¼ Y ðq0 ðti ÞÞQa ðti Þq_ ðtÞ þ ke ðqa0 ðnÞÞY ðqa0 ðnÞÞq_ t ðtÞq_ ðnÞ dn: ð40Þ b a 0 ti Let Zðqa0 Þ :¼
Z
qa0 0
Y ðpÞQ0a ðpÞ dp; bðpÞ
ð41Þ
such that 0
YQa a Z_ ¼ Z 0 q_ a0 ¼ q_ : b 0
ð42Þ
From the inequality (17) and Q0a > 0, b > 0, and Y > 0 in view of the definition (26), we have Z_ P 0;
Z0 ¼
YQ0a > 0; b
ð43Þ
which ensures the invertibility of the material function Zðqa0 Þ. Hence, qa0 ¼ qa0 ðZÞ
ð44Þ
is available, and all the material functions that originally expressed in terms of qa0 can be re-expressed in terms of Z. However, for saving notations we use the same symbols to denote such new functions, for example, Y ðZÞ :¼ Y ðqa0 ðZÞÞ:
ð45Þ
Substituting Eqs. (42) and (44) into Eq. (40), we obtain Z t Z_ ðtÞ ¼ Y ðZðti ÞÞQta ðti Þq_ ðtÞ þ ke ðZðnÞÞY ðZðnÞÞq_ t ðtÞq_ ðnÞ dn:
ð46Þ
ti
Then, integration gives ZðtÞ ¼ Zðti Þ þ Y ðZðti ÞÞQta ðti Þ½qðtÞ qðti Þ þ
Z
t
ke ðZðnÞÞY ðZðnÞÞ½qðtÞ qðnÞt q_ ðnÞ dn;
ti
which is a nonlinear Volterra integral equation for Z.
ð47Þ
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257
6. Homeomorphisms of measures of plastic irreversibility From Eqs. (26), (42), (43)1 and (24)2 it follows that bðke þ kp Þ _ Z P 0; Y_ ¼ 2 ðQ0a Þ
ð48Þ
and the dissipation power is K_ :¼ Qta q_ p ¼ Q0a q_ a0 P 0:
ð49Þ
In view of the switch (39), inequalities (17) and (43)1 , and Eqs. (42), (48) and (49), it is important to note that qa0 , Y , Z and K are intimately related in the sense that there exists strictly monotonic increasing relation between any pair of them as shown in Table 1. Therefore, any one of those irreversibility parameters can be chosen to play the role in the switch (39), the role of an indicator of irreversible change of material properties, the role of so-called material age, intrinsic time, internal time, the arrow of time, etc., in this time-independent model. Furthermore, they serve as the measures of plastic irreversibility and are crucially important in the evolutions of material properties and parameters and also in the determination of responses as shown in Eqs. (27), (28) and (31). Once Z is obtained, the other parameters are readily calculable through the integrals of the material homeomorphic functions listed in Table 1. Hence, Eq. (47) or Eq. (46) deserves a further study.
7. Material functions in terms of Z By Eqs. (41) and (44) the irreversibility parameter Z and the equivalent generalized plastic strain qa0 are closely related, being bijective, invertible, and strictly monotonic. This fact of equivalence can be utilized to accelerate the calculation of the material responses. For the generalized strain-controlled processes we may thereby change the dependence of the defined material functions on qa0 to directly on Z, and thus all the material functions with the arguments qa0 turn to the material functions simply with the arguments Z through Eq. (44). Even the process listed in the above is available to reveal the function dependence of Q0a ðZÞ, kp ðZÞ and so on. But they are not so straightforward. In this section the relationships of the material functions are further studied from a different point-of-view. Table 1 Relationships between irreversibility parameters q_ a Y_ 0
q_ a0
1
Q0a ðke þ kp ÞY
Y_
ðke þ kp ÞY Q0a
1
Z_ K_
Q0a Y b Q0a
X_ a0
ke Y b
X_ c0
ke Y ½1 2q0 Xc0 2 b½1 2q0 Q0a ð0Þ
ðQ0a Þ2 bðke þ kp Þ
Z_
K_
X_ a0
X_ c0
b Q0a Y
1 Q0a
b ke Y
ke Y ½1 2q0 Xc0 2
bðke þ kp Þ
ðke þ kp ÞY ðQ0a Þ2
bðke þ kp Þ ke Q0a
bðke þ kp Þ½1 2q0 Q0a ð0Þ
ðQ0a Þ2 Y b
Q0a ke
Q0a ½1 2q0 Q0a ð0Þ bQ0a ½1 2q0 Q0a ð0Þ
1
ðQ0a Þ2 ðke þ kp ÞY
b Y
1
bQ0a ke Y
ke Q0a
ke Q0a
ke Y bQ0a
1
ke ½1 2q0 Xc0 2 Q0a ½1 2q0 Q0a ð0Þ
ke Y ½1 2q0 Xc0 2 bQ0a ½1 2q0 Q0a ð0Þ
½1 2q0 Xc0 2 1 2q0 Q0a ð0Þ
bðke þ kp Þ 2q0 Xc0 2 kp Þ½1 2q0 Q0a ð0Þ
ke Q0a ½1 bðke þ
b½1 2q0 Q0a ð0Þ
ke Q0a ½1 2q0 Xc0 2 ke ½1 2q0 Xc0 2 ke Y ½1 2q0 Xc0 2 1 2q0 Q0a ð0Þ ½1 2q0 Xc0 2 1
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C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
In order to give the dependence of the material functions on Z directly, we combine Eqs. (45) and (26) to get Z qa
0 k ðpÞ þ k ðpÞ e p exp dp ¼ Y ðZÞ: ð50Þ Q0a ðpÞ 0 Derivating both sides with respect to qa0 , applying the chain rule on the right-hand side and then using Eq. (43)2 , we obtain bðqa0 Þ½ke ðqa0 Þ þ kp ðqa0 Þ ðQ0a ðqa0 ÞÞ
¼ Y 0 ðZÞ:
2
ð51Þ
Similarly, the derivative of Eq. (32) with respect to qa0 and the use of Eq. (43)2 give bðqa0 Þke ðqa0 Þ ðQ0a ðqa0 ÞÞ
2
¼ Y 2 ðZÞCe0 ðZÞ:
ð52Þ
By dividing the above two equations we obtain ke ðZÞY 0 ðZÞ ; Y 2 ðZÞCe0 ðZÞ
ð53Þ
Y 0 ðZÞ 1 : Y 2 ðZÞCe0 ðZÞ
ð54Þ
ke ðqa0 Þ þ kp ðqa0 Þ ¼ or in terms of Z, kp ðZÞ ¼ ke ðZÞ
The replacement of b in Eq. (52) by the one in Eq. (35) and the substitution of Eq. (53) into the resultant yield
ke2 ðZÞ ke ðZÞY 0 ðZÞ 0 a 2 00 a ¼ ðQa ðq0 ÞÞ Qa ðq0 Þ þ 2 : ð55Þ Y 2 ðZÞCe0 ðZÞ Y ðZÞCe0 ðZÞ 0
By the chain rule and the use of Eq. (43)2 the term Q0a ðqa0 Þ in the above equation can be written as 0
0
Q0a ðqa0 Þ ¼ Q0a ðZÞ
Y ðZÞQ0a ðZÞ : bðZÞ
ð56Þ
Upon using Eqs. (53) and (35) it can be arranged to 0
0 Q0a ðqa0 Þ
¼
0
ðZÞY ðZÞ Q0a ðZÞQ0a ðZÞ kYeðZÞC 0 ðZÞ e
ke ðZÞ Y ðZÞQ0a ðZÞQ0a0 ðZÞ
;
which is then substituted into Eq. (55), giving 2 3 ke ðZÞY 0 ðZÞ 0 00 0 Q ðZÞQ ðZÞ 0 ðZÞ ke2 ðZÞ k ðZÞY ðZÞ a a Y ðZÞC e 2 e 5: ¼ ðQ0a ðZÞÞ 4 þ Y 2 ðZÞCe0 ðZÞ ke ðZÞ Y ðZÞQ0a ðZÞQ0a0 ðZÞ Y 2 ðZÞCe0 ðZÞ
ð57Þ
ð58Þ
With some algebraic manipulations the following differential equation relating Y ðZÞ and ðQ0a ðZÞÞ2 can be obtained, d 2Y 0 ðZÞ 0 ke ðZÞ 2 2 ðQ0a ðZÞÞ þ ðQa ðZÞÞ ¼ : dZ Y ðZÞ Y ðZÞ
ð59Þ
This equation provides an interesting relation among the three material functions ke ðZÞ, Q0a ðZÞ and Y ðZÞ.
C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
259
2
If Y ðZÞ and ke ðZÞ are given then the solution of ðQ0a ðZÞÞ obtained from Eq. (59) is Z Z 2 2 ðQ0 ð0ÞÞ 2 ðQ0a ðZÞÞ ¼ 2 ; ke ðpÞY ðpÞ dp þ a2 Y ðZÞ 0 Y ðZÞ and, if Q0a ðZÞ and ke ðZÞ are given then the solution of Y ðZÞ obtained from Eq. (59) is Z Z 1 ke ðpÞ Q0a ð0Þ dp þ : Y ðZÞ ¼ 0 Qa ðZÞ 0 Q0a ðpÞ Q0a ðZÞ Ce ðZÞ is calculated from Eq. (54) by Z Z ke ðpÞY 0 ðpÞ dp; Ce ðZÞ ¼ 2 0 ½ke ðpÞ þ kp ðpÞY ðpÞ
ð60Þ
ð61Þ
ð62Þ
and kp ðZÞ is evaluated by Eq. (54). Similarly, Z Z 0 Ce ðpÞ dp CðZÞ :¼ ke ðpÞ 0
ð63Þ
is obtained from Eq. (29) by changing the integration variable by means of Eqs. (42) and (52). 2 Eq. (60) provides a functional relation between ðYQ0a Þ and ke Y , and Eq. (61) a functional relation between YQ0a and ke =Q0a . The above two equations are consistent. Interestingly, the parameter YQ0a has already appeared in several places, for example, Eqs. (29), (32), (41)–(43), etc. In Section 9 we will reveal the importance of the parameter YQ0a for further understanding of the mixed-hardening plasticity model.
8. Operators in terms of Z Once Z is obtained, the material functions Q0a ðZÞ and kp ðZÞ are specified, and Y ðZÞ and Ce ðZÞ are evaluated via Eqs. (61) and (62) (or Y ðZÞ and Ce ðZÞ are specified, and Q0a ðZÞ and kp ðZÞ are evaluated via Eqs. (60) and (54)), Qa ðtÞ and QðtÞ can be calculated respectively by
Z t 1 Y ðZðti ÞÞQa ðti Þ þ Qa ðtÞ ¼ ke ðZðnÞÞY ðZðnÞÞq_ ðnÞ dn ; ð64Þ Y ðZðtÞÞ ti QðtÞ ¼ Qðti Þ þ ½Gs ðZðtÞ; Zðti ÞÞ Gs ðZðti Þ; Zðti ÞÞ
Qa ðti Þ þ ke ðZðti ÞÞ
Z
t
Gs ðZðtÞ; ZðnÞÞq_ ðnÞ dn;
ð65Þ
ti
where Gs ðZ1 ; Z2 Þ :¼ ke ðZ2 Þf1 Y ðZ2 Þ½Ce ðZ1 Þ Ce ðZ2 Þg: Similarly, the generalized plastic strain is calculated by Z t Gp ðZðtÞ; Zðti ÞÞ qp ðtÞ ¼ qp ðti Þ þ Qa ðti Þ þ Gp ðZðtÞ; ZðnÞÞq_ ðnÞ dn; ke ðZðti ÞÞ ti
ð66Þ
ð67Þ
where Gp ðZ1 ; Z2 Þ :¼ ke ðZ2 ÞY ðZ2 Þ½CðZ1 Þ CðZ2 Þ:
ð68Þ
A numerical procedure based on the discretizations of Eqs. (47), (64), (65) and (67) can be developed to calculate the responses of mixed-hardening elastoplastic model under general loading conditions. However, this important computational issue is reported by Liu (submitted for publication) in other place.
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9. Augmented differential equations system From Eqs. (46) and (64) we have
1
Z_ ¼ Y ðZÞQta q_ ;
ð69Þ
and from Eqs. (25), (26), (44) and (45) we have d ½Y ðZÞQa ¼ ke ðZÞY ðZÞq_ ; dt
ð70Þ
which can be rearranged to d ke ðZÞ ½Y ðZÞQa ¼ Y ðZÞQ0a ðZÞ 0 q_ : dt Qa ðZÞ
ð71Þ
On the other hand, multiplying Eq. (61) by Q0a on both sides, taking the time derivative and then using Eq. (69) we obtain d ke ðZÞ t ½Y ðZÞQ0a ðZÞ ¼ Y ðZÞ 0 Q q_ : dt Qa ðZÞ a Let us introduce s
Xa Y Qa Xa ¼ :¼ Xa0 YQ0a
ð72Þ
ð73Þ
as the augmented (n þ 1)-dimensional state vector, 2 where Xa0 is another irreversible parameter, and Table 1 shows its relations with the other irreversible parameters. Correspondingly, Xsa :¼ Y Qa is the spatial part of Xa in the Minkowski spacetime, and we may call it the augmented generalized active stress. Moreover, from Eqs. (39), (72) and (73) it follows that ( 0 ke Xa Qt q_ > 0 if kQa k ¼ Q0a and Qta q_ > 0; X_ a0 ¼ ðQ0a Þ2 a ð74Þ 0 if kQa k < Q0a or Qta q_ 6 0: Thus, Eqs. (71) and (72) can be combined together as following equations system: X_ a ¼ AXa ;
ð75Þ
where A :¼
ke 0nn Q0a q_ t
q_ : 0
Define the proper time s as follows: Z t ke ke ðnÞ dn; ds ¼ 0 dt; s ¼ si þ 0 ðnÞ Q Qa ti a
ð76Þ
ð77Þ
1 Because Y ðZÞQa is the augmented generalized active stress, we may correspondingly call Z_ the augmented generalized active power. 2 From Eqs. (73) and (61) we have X_ a0 ¼ ke Z_ =Q0a > 0 in the on phase, and thus Xa0 is a time-like parameter which can be viewed as the temporal component of Xa in the Minkowski spacetime.
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where ti is an initial time and si is the corresponding initial proper time. Since the term ke =Q0a in Eq. (76) is positive, the proper time s is a monotonic function of the external time t. The on-off switching criteria in Eq. (74) can thus be written as dXa0 ¼ ds
(
Xa0 Q0a
Qta q_ > 0 if kQa k ¼ Q0a and Qta q_ > 0; 0 if kQa k < Q0a or Qta q_ 6 0:
ð78Þ
In terms of the new time scale s, Eq. (75) becomes d Xa ¼ BXa ; ds
ð79Þ
where
0 B :¼ nn q_ t
q_ : 0
ð80Þ
It can be seen that in this augmented space Xa ¼ ðY Qta ; YQ0a Þt , the governing equations become linear with respect to s and are more tractable than the original nonlinear equations. From Eqs. (79) and (80) a deeper understanding of the underlying structure of the model may be achieved as to be done in Sections 10–12.
10. The Minkowski spacetime In Section 9 we found that even the constitutive equations are nonlinear in the n-dimensional state space of generalized active stresses Qa , but they can be transformed to linear differential equations in the (n þ 1)dimensional augmented state space of Xa through a time scaling. In the augmented space not only the nonlinearity of the model is unfolded, but also an intrinsic spacetime structure of the Minkowskian type will be brought out. Now we translate the mixed-hardening elastoplastic model postulated in Section 1 in the state space of Qa to one in the augmented state space of Xa . Accordingly the first row in Eq. (79), and Eqs. (18), (16) and (17) become
In 0n1 dXa 0nn q_ ¼ ð81Þ X; 01n Xta gXa ds 01n 0 a Xta gXa 6 0;
ð82Þ
dXa0 P 0; ds
ð83Þ
in terms of the Minkowski metric (in the space-like convention)
In 0n1 g :¼ ; 01n 1
ð84Þ
where In is the identity matrix of order n. The vector space of augmented states Xa endowed with the Minkowski metric tensor g is referred to as Minkowski spacetime, and designated as Mnþ1 . Regarding Eqs. (16) and (82), we further distinguish two correspondences: kQa k ¼ Q0a () Xta gXa ¼ 0;
ð85Þ
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kQa k < Q0a () Xta gXa < 0:
ð86Þ
That is, a generalized active stress state Qa on (resp. within) the yield hypersphere kQa k ¼ Q0a in the generalized active stress space of (Q1a ; . . . ; Qna ) corresponds to an augmented state Xa on the cone fXa jXta gXa ¼ 0g of Minkowski spacetime (resp. in the interior fXa jXta gXa < 0g of the cone). The exterior fXa jXta gXa > 0g of the cone is uninhabitable since kQa k > Q0a is forbidden. Even though it admits an infinite number of Riemannian metrics, the yield hypersphere Sn1 of Qa does not admit a Minkowskian metric, nor does the space of (Qta ; Y ). It is the cone of Xa which admits the Minkowski metric. From Eqs. (34), (73) and (84) it follows that t
kQa k ¼ Q0a ) Xta gðq_ t ; q_ a0 =bÞ ¼ 0:
ð87Þ
Moreover, by Eqs. (73), (78)1 , (79) and (84) we can prove that fkQa k ¼ Q0a and Qta q_ > 0g ) Xta g
dXa ¼ 0: ds
ð88Þ
If the model is in the on phase (i.e., not only kQa k ¼ Q0a but also Qta q_ > 0), then from Eqs. (85), (87) and (88) we assert that for an Xa -path on the cone, the augmented state Xa is M-orthogonal to itself, its tangent vector dXa =ds and also its dual ðq_ t ; q_ a0 =bÞ. The so called M-orthogonality is an orthogonality of two (n þ 1)vectors with respect to metric (84) in Minkowski spacetime Mnþ1 (see, e.g., Das, 1993 and Naber, 1992). On the other hand, Xa0 is frozen in the off phase as indicated by Eq. (78)2 and the augmented state Xa stays in the closed n-disc Dn (i.e., closed n-ball Bn ) on the hyperplane Xa0 ¼ constant in the space of (Xa1 ; . . . ; Xan ; Xa0 ), the hyperplane being identified to be Euclidean n-space En , which is endowed with the Euclidean metric In . In summary, the augmented state Xa either evolves on the cone when in the on phase or stays in the discs of simultaneity, which are stacked up one by one in the interior of the cone and are glued to the cone, when in the off phase.
11. Space-like paths in the Minkowski spacetime The criteria in Eq. (39) ensure that _ a Q0 Q00 q_ a ¼ 0 q_ a0 ½Qta Q a a 0
ð89Þ
no matter whether in the on or in the off phase. Substituting Eqs. (11)–(15) into the above equation and using Eq. (23), we obtain bq_ t q_ p ¼ ðq_ p Þt q_ p P 0:
ð90Þ
Materials which satisfy such an inequality q_ t q_ p P 0 are said to be kinematically stable, e.g., Lubliner (1984). Using Eq. (23) and the above equation we obtain pffiffiffiffiffiffiffiffiffiffiffi ð91Þ q_ a0 ¼ bq_ t q_ p : Squaring of the above equation and using Eqs. (17) and (14), we get 2
ðq_ a0 Þ ¼ q_ a0
bQta q_ ; Q0a
ð92Þ
by which we have 2
ðq_ a0 Þ 6
bq_ a0 kQa kkq_ k; Q0a
ð93Þ
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and using Eq. (18) the bound of q_ a0 can be derived, q_ a0 6 bkq_ k
ð94Þ
no matter in the on or off phase. This inequality tells us that the maximum value of the specific dissipation power K_ ¼ Q0a q_ a0 an admissible path in the state space may discharge is bQ0a kq_ k. (On the other hand, postulation (17) tells us that the minimum value of the specific dissipation power an admissible path may discharge is zero.) The solution of Eq. (79) may be viewed as a path in Mnþ1 , which traces along the null cone Xta gXa ¼ 0 given by Eq. (85). From which the derivative gives Xta g dXa ¼ 0:
ð95Þ
Thus Xa is M-orthogonal to dXa . Furthermore, upon using this equation and 2 !2 3 n n n n X X X X 1 4 2 2 2 2 ðdXai Þ ðdXa0 Þ ¼ ðXai Þ ðdXai Þ Xai dXai 5: 0 Þ2 ðX i¼1 i¼1 i¼1 i¼1 a
Xta gXa
¼ 0 we have ð96Þ
So that by utilizing the Schwartz inequality we obtain 2
t
ðdXa Þ :¼ ðdXa Þ g dXa P 0:
ð97Þ t
Recalling that a path such that ðdXa Þ g dXa > 0 (resp: ¼ 0; < 0) is called a space-like (resp. null, timelike) path in Mnþ1 , we thereby conclude that the path Xa ðs0 Þ, si < s0 6 s, in the augmented state space is a space-like or null path in the Minkowski spacetime Mnþ1 no matter in the on or in the off phase. Indeed, Eq. (97) conveys a message that the nature of mixed-hardening elastoplasticity rejects time-like paths, so that the time-like metric convention has to be rejected to avoid an unreasonable negative squared length. This is the reason why we have adopted the space-like convention (84) for mixed-hardening elastoplasticity.
12. The proper orthochronous Lorentz group In this section we concentrate on the on phase to bring out internal symmetry inherent in the model in the on phase. Denote by Ion an open, maximal, continuous proper time interval during which the mechanism of plasticity is on exclusively. The solution of the augmented state equation (79) can be expressed as in the following augmented state transition formula: Xa ðsÞ ¼ ½GðsÞG1 ðs1 ÞXa ðs1 Þ;
8 s; s1 2 Ion ;
ð98Þ
in which GðsÞ, called the fundamental matrix of Eq. (79), is a square matrix of order n þ 1 satisfying d GðsÞ ¼ BðsÞGðsÞ; ds Gð0Þ ¼ Inþ1 :
ð99Þ ð100Þ
On the other hand, from Eqs. (80) and (84) it is easy to verify that the control matrix B in the on phase satisfies Bt g þ gB ¼ 0: By Eqs. (101) and (99) we find d t ½G ðsÞgGðsÞ ¼ 0: ds
ð101Þ
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From Eq. (100) we have Gt ðsÞgGðsÞ ¼ Itnþ1 gInþ1 ¼ g at s ¼ 0, and thus prove that Gt ðsÞgGðsÞ ¼ g
ð102Þ
for all s 2 Ion . Take determinants of both sides of the above equation, getting 2
ðdet GÞ ¼ 1;
ð103Þ
so that G is invertible. The 00 entry of the matrix equation (102) is ðG00 Þ2 P 1:
Pn
i 2 i¼1 ðG0 Þ
2
ðG00 Þ ¼ 1, from which ð104Þ
Here Gij ; i; j ¼ 1; . . . ; n; 0; is the mixed ij-entry of the matrix G. Since det G ¼ 1 or G00 6 1 would violate Eq. (100), it turns out that det G ¼ 1;
ð105Þ
G00 P 1:
ð106Þ
In summary, G has the three characteristic properties explicitly expressed by Eqs. (102), (105) and (106). Recall that the complete homogeneous Lorentz group Oðn; 1Þ is the group of all invertible linear transformations in Minkowski spacetime which leave the Minkowski metric invariant, and that the proper orthochronous Lorentz group SOo ðn; 1Þ is a subgroup of Oðn; 1Þ in which the transformations are proper (i.e., orientation preserving, namely the determinants of the transformations being þ1) and orthochronous (i.e., time-orientation preserving, namely the 00 entry of the matrix representations of the transformations being positive) (see, e.g., Cornwell, 1984). Hence, in view of the three characteristic properties we conclude that the fundamental matrix G belongs to the proper orthochronous Lorentz group SOo ðn; 1Þ. So the matrix function GðsÞ of proper time s 2 Ion may be viewed as a connected path of the Lorentz group. Furthermore, by Eq. (101), B is an element of the real Lie algebra soðn; 1Þ of the Lorentz group SOo ðn; 1Þ. From Eq. (78)1 , dXa0 =ds > 0 strictly when the mechanism of plasticity is on; hence, 3 Xa0 ðsÞ > Xa0 ðs1 Þ > Xa0 ðs0 Þ ¼ Q0a ;
8s > s1 > s0 ;
s; s1 2 Ion ;
ð107Þ
which means that in the sense of irreversibility there exists future-pointing proper time-orientation from the augmented states Xa ðs1 Þ to Xa ðsÞ. Moreover, such time-orientation is a causal one, because the augmented state transition formula (98) and inequality (107) establish a causality relation between the two augmented states Xa ðs1 Þ and Xa ðsÞ in the sense that the preceding augmented state Xa ðs1 Þ influences the following augmented state Xa ðsÞ according to formula (98). Accordingly, the augmented state Xa ðs1 Þ chronologically and causally precedes the augmented state Xa ðsÞ. This is indeed a common property for all models with inherent symmetry of the proper orthochronous Lorentz group. By this symmetry a core connection among irreversibility, the time arrow of evolution, and causality has thus been established for plasticity in the on phase.
13. The Poincare group The proper orthochronous Lorentz group SOo ðn; 1Þ constructed in the above is acting on the space Xa ; and what is the effect of the kinematic hardening on the group structure? In order to reply this question we 3 From Eqs. (17), (26) and (73) it follows that Xa0 ðsÞ P Xa0 ðs0 Þ P Xa0 ðsi Þ P Xa0 ðs0 Þ ¼ Q0a for all s P s0 P si P s0 , applicable to both the on and off phases. Recall that s0 is the zero-value proper time at which all relevant values including qa0 ðs0 Þ ¼ 0.
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return to the derivation of Eq. (31) as follows. First, from Eqs. (15), (14) and (27), the generalized back stress is integrated as follows: Z t Gb ðqa0 ðtÞ; qa0 ðti ÞÞ Qa ðti Þ þ Qb ðtÞ ¼ Qb ðti Þ þ Gb ðqa0 ðtÞ; qa0 ðnÞÞq_ ðnÞ dn; ð108Þ ke ðqa0 ðti ÞÞ ti where Z
qa0
kp ðpÞ dp; Y ðpÞQ0a ðpÞ
ð109Þ
Gb ðp1 ; p2 Þ :¼ ke ðp2 ÞY ðp2 Þ Cp ðp1 Þ Cp ðp2 Þ :
ð110Þ
Cp ðqa0 Þ :¼
0
Both Cp and Gb are material functions. In terms of Ga ðp1 ; p2 Þ :¼
ke ðp2 ÞY ðp2 Þ ; Y ðp1 Þ
ð111Þ
it can be verified from Eqs. (26), (32), (33), (110) and (109) that Gs ðp1 ; p2 Þ :¼ ke ðp2 Þf1 Y ðp2 Þ½Ce ðp1 Þ Ce ðp2 Þg ¼ Ga ðp1 ; p2 Þ þ Gb ðp1 ; p2 Þ:
ð112Þ
Combining Eqs. (27) and (108) and noting Eqs. (12) and (112) yield Eq. (31) again. Equation (108) is an integral representation expressing the generalized back stress in terms of the generalized strain rate history. Now, introduce the space X with the following vector decomposition:
Y Qa Qb X ¼ Xa þ Xb :¼ : ð113Þ þ YQ0a 0 The group acting on the space X is the semi-direct product of the translation Tnþ1 and the proper orthochronous Lorentz group SOo ðn; 1Þ; usually such group is named the Poincare group ISOo ðn; 1Þ, or inhomogeneous Lorentz group (see, e.g., Kim and Noz, 1986).
14. Conformal spacetime From Eqs. (11)–(13), (15) and (91) we have 2
2
0
2
2
ðdQa Þ :¼ kdQa k ¼ ke2 kdqk ðke þ kp Þ½ke þ kp þ 2Q0a ðdqa0 Þ ;
ð114Þ
which, with the aid of Eqs. (71)–(73), (39), (84) and (97), leads to 0
ðdXa Þ2 ¼ Y 2 ½ðdQa Þ2 ðQ0a Þ2 ðdqa0 Þ2
ð115Þ 00
no matter in the on or in the off phase. Especially, in the on phase we have Qa dQa ¼ Q0a Qa dqa0 by the 0 plastic consistency condition, and for the nonperfect case, i.e., Q0a 6¼ 0, we have " # " # 2 ðQ dQ Þ 1 0 2 2 2 2 2 a a ðdQa Þ ðdqa0 Þ : ðdXa Þ ¼ Y 2 kdQa k ¼ Y 2 ðQ0a Þ ð116Þ 2 2 ðQ0a Þ ðQ0a0 Þ 2
0
2
2
The metric line element ðdQa Þ =ðQ0a Þ ðdqa0 Þ defined in the space (Qta ; qa0 ) indicates that the spatial sec0 tions expand (or contract) uniformly as described by the scalar function Q0a ðqa0 Þ. This form of the metric line element is manifestly conformal to the Minkowski space Xa with a conformal scalar factor Y ðqa0 Þ. The space (Qta ; qa0 ) is known as a Robertson–Walker spacetime (see, e.g., Hawking and Ellis, 1973 and Birrell and Davies, 1982).
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In this occasion we would like to point out that the metric considered by De Saxce and Hung (1985), who attempted to study plasticity models from a differential geometry view, is not a suitable one. Really, by 2 identifying ðQ0a Þ ¼ 2 r2 =3, dK ¼ Q0a dqa0 ¼ dj0 and dqpa ¼ dja with the notations used by De Saxce and Hung, we readily obtain 2
dr2 ¼ ðdj0 Þ
2 r2 dab dja djb 3
from the flow rule (14). See also Eq. (5.2) proposed by De Saxce and Hung, loc.cit. The above metric vanishes no matter whether in the on or in the off phase. In this sense such metric is not a suitable measure for its full degeneracy. More precisely, it is not a metric and also not a hyperbolic metric. So, De Saxce and Hung asserting it as a hyperbolic metric on the base manifold is incorrect. Indeed, dr2 ¼ 0 is at most an identity derived from the associated flow rule. t Eq. (75) is the governing equation of Xa ¼ ðY Qta ; YQ0a Þ ; and from Eqs. (71), (72), (26), (77) and (39) the governing equation of ðQa ; Q0a Þ is read as
d Qa ke 0nn q_ Qa bðke þ kp Þ t Qa _ Qa q 0 : ¼ 0 ð117Þ 2 Qa 0 Q0a dt Q0a Qa q_ t ðQ0a Þ t
However, ðQta ; Q0a Þ is not a suitable spacetime structure, since dQ0a may be negative in the softening range of modeled material. In order to obtain a suitable conformal spacetime structure, we introduce a scaling factor function qðqa0 Þ and consider the following augmented vector: s
Xc qQa Xc ¼ :¼ ; ð118Þ Xc0 qQ0a in which we need dðqQ0a Þ > 0 in the on phase, and simultaneously the cone is preserved, i.e., ðXc Þt gXc ¼ 0:
ð119Þ
Taking the time derivative of Eq. (118) and using (117) we obtain
d qQa ke 0nn q_ qQa bðke þ kp Þ q_ qQa qQa t _ q qQ ¼ þ : a qQ0a 0 qQ0a dt qQ0a q qQ0a Q0a q_ t qðQ0 Þ2
ð120Þ
a
If q satisfies
ke bðke þ kp Þ q_ t _; ¼ 0 2q 0 qQa q qke Q0a q Qa
ð121Þ
where q0 is a constant whose range to be determined below, then in the space Xc the governing equations are given by
d qQa ke 0nn q_ qQa 2ke q0 qQa t _ q qQ ¼ : ð122Þ a qQ0a 0 qQ0a dt qQ0a Q0a q_ t Q0a By using the proper time
d qQa 0 ¼ nn q_ t ds qQ0a
s defined in Eq. (77) the above equations are further refined to
qQa q_ qQa t _ q qQ 2q : 0 a qQ0a 0 qQ0a
ð123Þ
The above differential equations system is one of the conformal type, e.g., Anderson et al. (1982), with the (n þ 1)-dimensional conformal parameters (q0 q_ ; 0). The group generated from the above differential equations system is a special type of conformal group.
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Eq. (121), after substituting q_ ¼ q0 q_ a0 and Eq. (39)1 for Qta q_ ¼ Q0a q_ a0 =b and cancelling q_ a0 Õs on both sides, can be changed to the following Riccati differential equation: q0 ¼
ke þ kp 2ke q0 2 q: q Q0a b
ð124Þ
Using Eq. (26) the solution of q is found to be qðqa0 Þ ¼
1 þ 2q0
Y ðqa0 Þ R qa0 ke ðpÞY ðpÞ 0
bðpÞ
ð125Þ
: dp
On the other hand, from Eqs. (61) and (42) we have dðYQ0a Þ ke _ ke Y a ¼ 0Z ¼ q_ > 0 dt b 0 Qa
ð126Þ
in the on phase, such that the integral in Eq. (125) can be obtained explicitly, qðqa0 Þ ¼
Y ðqa0 Þ : 1 2q0 Q0a ð0Þ þ 2q0 Y ðqa0 ÞQ0a ðqa0 Þ
ð127Þ
When q0 ¼ 0 we have q ¼ Y , and the differential equations system (79) is recovered from Eq. (123). We also note that q0 can not be a negative constant value; otherwise, the above q will blow-up when Y increases to a certain positive value. From Eq. (127) the relation of Xa and Xc through Eqs. (73) and (118) is available, Xc ¼
1 Xa : 1 2q0 Q0a ð0Þ þ 2q0 Xa0
ð128Þ
Furthermore, the metrics in both the spaces Xc and Xa can be proved to be related conformally as follows: 2
ðdXc Þ ¼
½1 2q0 Xc0 ½1
2
1
2
2 2q0 Q0a ð0Þ
ðdXa Þ ¼
½1
2
2q0 Q0a ð0Þ
2
þ 2q0 Xa0
ðdXa Þ ;
ð129Þ
where 2
t
ðdXc Þ :¼ ðdXc Þ g dXc :
ð130Þ
Finally, from Eq. (128) it is easy to prove that X_ c0 ¼
½1 2q0 Q0a ð0Þ ½1 2q0 Q0a ð0Þ þ
2q0 Xa0 2
2
½1 2q0 Xc0 _ 0 X : X_ a0 ¼ 1 2q0 Q0a ð0Þ a
ð131Þ
For the purpose of Xc0 > 0 to be an irreversible parameter as well, i.e., X_ c0 > 0, by Eqs. (126) and (73) we require 1 2q0 Q0a ð0Þ > 0, that is, 0 6 q0 <
1 : 2Q0a ð0Þ
ð132Þ
Thus, under this condition we have introduced another irreversible parameter Xc0 as the temporal component of the conformal spacetime, the relations of which to the other irreversible parameters are listed in Table 1. It includes two dimensionless irreversible parameters qa0 and Y , and also four stress dimension irreversible parameters Z, K, Xa0 and Xc0 . The hyperbolic geometric models that we studied for the mixed-hardening elastoplasticity are governed by the conformally isormorphic spacetimes. They are all differentiable manifold endowed with a
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pseudo-Riemann metric, which assigns at each spacetime point Xc an indefinite symmetric inner product on the tangent space, which varying differentiably with Xc .
15. Conclusions In this paper we have established two types formulations for mixed-hardening elastoplastic model of the generalized stress and generalized strain: the flow model characterized by Eqs. (11)–(18) and the two-phase linear differential system characterized by Eqs. (79) and (80). Even though the governing equations of the generalized active stress are nonlinear in the n-dimensional space Qa we have found that it can be transformed to linear differential equations in the augmented (n þ 1)-dimensional space Xa . In this space not only the nonlinearity of the model can be unfolded, but also an intrinsic spacetime structure of the Minkowskian type can be revealed, merely replacing the inequality in space Qa to the inequality in the augmented space Xa . We also pointed out that the state matrix B is an element of the Lie algebra of the proper orthochronous Lorentz group, hence the state transition matrix generated from the linear differential equations was proved to be a type of the proper orthochronous Lorentz group. In the frame of the Minkowski spacetime we have further proved that the action of the kinematic rule in the mixed-hardening model causes a translation of the state X, which amounts to extend the proper orthochronous Lorentz group to the proper orthochronous Poincare group. We have introduced a conformal factor q in Eq. (127) and thus a conformal spacetime Xc is derived. The analytic models of hyperbolic geometry that we studied for the mixed-hardening elastoplasticity are governed by the conformally isormorphic spacetimes. They are all differentiable manifold endowed with a pseudo-Riemann metric, which assigns at each spacetime point an indefinite symmetric inner product on the tangent space, which varying differentiably. Mathematically speaking, the pseudo-Riemannian manifold is a suitable underlying spacetime model for the mixed-hardening elastoplasticity. Even we dealt only with mixed-hardening effect on the elastoplastic model without considering a more inclusive Armstrong-Frederick kinematic hardening rule, 4 and/or large deformation, etc.; however, we may further consider more sophisticated group actions on pseudo-Riemannian manifolds and thus makes an explicit use of the powerful group-theoretic method to study plasticity from a global view. In this regard, the present paper may open the way to plasticity research in a new direction.
Acknowledgement The financial support provided by the National Science Council under Grant NSC 90-2212-E-019-007 is gratefully acknowledged.
References Anderson, R.L., Harnad, J., Winternitz, P., 1982. Systems of ordinary differential equations with nonlinear superposition principles. Physica 4D, 164–182. Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. G.E.G.B. Report RD/B/N 731. Birrell, N., Davies, P., 1982. Quantum Fields in Curved Space. Cambridge University Press, Cambridge.
4 _ b ¼ kp q_ p kb q_ a Qb with kb > 0 to account Armstrong and Frederick (1966) have proposed a nonlinear kinematic hardening rule Q 0 of the saturation of Qb under proportional loading conditions.
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Cornwell, J.F., 1984. In: Group Theory in Physics, vol. 2. Academic Press, London. Das, A., 1993. The Special Theory of Relativity. Springer-Verlag, New York. De Saxce, G., Hung, N.D., 1985. The geometric nature of plasticity laws. Engrg. Fract. Mech. 21, 781–798. Hawking, S.W., Ellis, G.F., 1973. The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge. Hong, H.-K., Liu, C.-S., 1993. Reconstructing J2 flow model for elastoplastic materials. Bull. College Engrg. N.T.U. 57, 95–114. Hong, H.-K., Liu, C.-S., 1999a. Lorentz group SOo ð5; 1Þ for perfect elastoplasticity with large deformation and a consistency numerical scheme. Int. J. Non-Linear Mech. 34, 1113–1130. Hong, H.-K., Liu, C.-S., 1999b. Internal symmetry in bilinear elastoplasticity. Int. J. Non-Linear Mech. 34, 279–288. Hong, H.-K., Liu, C.-S., 2000. Internal symmetry in the constitutive model of perfect elastoplasticity. Int. J. Non-Linear Mech. 35, 447–466. IlÕyushin, A.A., 1963. Plasticity: Foundation of General Mathematical Theory. Acad. Nauk SSSR, Moscow. Kim, Y.S., Noz, M.E., 1986. Theory and Applications of the Poincare Group. D. Reidel Publishing Company, Dordrecht. Liu, C.-S., 2001. The g-based Jordan algebra and Lie algebra with application to the model of visco-elastoplasticity. J. Marine Sci. Tech. 9, 1–13. Liu, C.-S., 2002. Applications of the Jordan and Lie algebras for some dynamical systems having internal symmetries. Int. J. Appl. Math. 8, 209–240. Liu, C.-S., submitted for publication. Lie symmetries of finite strain elastic-perfectly plastic models and exactly consistent schemes for numerical integrations. Liu, C.-S., submitted for publication. A consistent numerical scheme for the von Mises mixed-hardening constitutive equations. Lubliner, J., 1984. A maximum-dissipation principle in generalized plasticity. Acta Mech. 52, 225–237. Naber, G.L., 1992. The Geometry of Minkowski Spacetime. Springer-Verlag, New York.
Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity Chein-Shan Liu
*
Department of Mechanical and Marine Engineering, National Taiwan Ocean University, Keelung 202-24, Taiwan Received 24 May 2002; received in revised form 25 September 2002
Abstract The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which Xa0 and Xc0 play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector Xa :¼ ðY Qta ; YQ0a Þt is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime Mnþ1 on which the proper orthochronous Lorentz group SOo ðn; 1Þ left acts. Then, constructed is a Poincare group ISOo ðn; 1Þ on space X :¼ Xa þ Xb , of which Xb reflects the kinematic hardening rule in the model. We also find that the space ðQta ; qa0 Þ is a Robertson–Walker spacetime, which is conformal to Xa through a factor Y , and conformal to Xc :¼ ðqQta ; qQ0a Þt through a factor q as given by qðqa0 Þ ¼ Y ðqa0 Þ=½1 2q0 Q0a ð0Þ þ 2q0 Y ðqa0 ÞQ0a ðqa0 Þ. In the conformal spacetime the internal symmetry is a conformal group. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elastoplasticity; Mixed-hardening; Symmetry group; Minkowski spacetime; Conformal spacetime
1. Introduction The formulation of yield criteria of Tresca and of von Mises was the first key step in developing plasticity theory. However, without a plastic stress-strain relation, the criteria can do little. The incremental stressstrain relations proposed by Saint-Venant and Levy represented a giant step forward in plasticity theory. Since then the incremental or rate type point-of-view has been most frequently taken in formulating constitutive equations of plasticity, which describe the evolutions of internal state variables. This kind of rate-equation representations is now composed of separately specified but interwoven ingredients, including the elastic part, yield condition, loading-unloading criterion, plastic flow rule, isotropic hardening rule and kinematic hardening rule, etc. Because of the nonlinear nature of the model ingredients in plasticity, there seems necessary although difficult by composing all ingredients together to derive a global theory of plasticity, which not only gives
*
Tel.: +886-2-2462-2192x3252; fax: +886-2-2462-0836. E-mail address: [email protected] (C.-S. Liu).
0020-7683/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 2 ) 0 0 5 5 2 - 8
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directly a local response of the material in the time domain once an input is prescribed, but also facilitates us to handle model property from a global view. To do this several thresholds need to be stridden across. The first is integrating the rate-type equations, which consist of separated but interwoven ingredients specified on a high-dimensional space such as states in stress space and paths in plastic strain space and their product space. The second is solving the irreversibility parameters, such as the equivalent plastic strain, dissipation, and so on, and in general there exist monotonic mappings between these parameters. The third as to be strongly emphasized here is studying symmetry properties of the model and its intrinsic spacetime structures. The differential nature of plasticity laws has been discussed for a long time. The invariant yield condition in stress space renders a natural mathematical frame of plasticity theory from the view of differentiable manifold and its Lie group transformation. In this paper we are going to make an effort to study the internal symmetry groups and the underlying spacetime structures for mixed-hardening elastoplasticity. The Lie group theory provides a universal tool for tackling considerable numbers of differential equations when other means fail. Indeed, group analyses may augment intuition in understanding and in using symmetry for formulation of physical models, and often disclose possible approaches to solving complex problems. Recently, Hong and Liu (1999a,b, 2000) have considered constitutive models of perfect elastoplasticity with and without considering large deformation and of bilinear elastoplasticity, revealing that the models possess two kinds of internal symmetries, characterized by the translation group T ðnÞ in the off (or elastic) phase and by the projective proper orthochronous Lorentz group PSOo ðn; 1Þ in the on (or elastoplastic) phase for the first two models and the proper orthochronous Poincare group PISOo ðn; 1Þ in the on phase for the bilinear model, and have symmetry switching between the two groups dictated by the control input. Moreover, Liu (2001, 2002) applied a PDSOo ðn; 1Þ symmetry to formulate a mathematical model of visco-elastoplasticity, and then Liu (submitted for publication) employed a two-component spinor representation to unify finite strain elastic-perfectly plastic models with different objective stress rates and showed that the Lie symmetry in the two-dimensional spinor space is SLð2; HÞ with H denoting the quaternionic number. In this paper some of those results are extended to the model of mixed-hardening elastoplasticity, investigating the influence of nonlinear hardening parameters on the structure of the underlying vector spaces and on the transformation groups of internal symmetries.
2. Constitutive postulations In terms of five-dimensional stress vectors and five-dimensional strain vectors: 3 3 3 2 11 2 11 2 11 a1 sa þ a2 s22 a1 sb þ a2 s22 a1 s þ a2 s22 a b 6 a3 s11 þ a4 s22 7 6 a3 s11 þ a4 s22 7 6 a3 s11 þ a4 s22 7 7 6 6 a 23 a 7 6 b 23 b 7 23 7 7 7; 6 6 ; Q ; Q Q¼6 ¼ ¼ s sb s a b a 7 7 7 6 6 6 13 13 13 5 5 5 4 4 4 s s s a b 12 s12 s s12 a b 2
3 a1 e11 þ a2 e22 6 a3 e11 þ a4 e22 7 6 7 7; q¼6 e23 6 7 4 5 e13 e12
2
3 a1 ee11 þ a2 ee22 6 a3 ee11 þ a4 ee22 7 6 7 e 7; ee23 q ¼6 6 7 4 5 ee13 ee12
3 a1 ep11 þ a2 ep22 6 a3 ep11 þ a4 ep22 7 6 7 p 7; q ¼6 ep23 6 7 4 5 ep13 ep12
ð1Þ
2
ð2Þ
the tensorial form of elastoplastic model with mixed-hardening rule, e.g., Hong and Liu (1993): e_ ¼ e_ e þ e_ p ;
ð3Þ
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253
s ¼ sa þ sb ;
ð4Þ
s_ ¼ 2G_ee ;
ð5Þ
k_sa ¼ 2s0y e_ p ;
ð6Þ
s_ b ¼ 2k 0 e_ p ;
ð7Þ
pffiffiffi ksa k 6 2s0y ;
ð8Þ
k_ P 0;
ð9Þ
pffiffiffi ksa kk_ ¼ 2s0y k_;
ð10Þ
can be equivalently re-expressed by the following vectorial form: q_ ¼ q_ e þ q_ p ;
ð11Þ
Q ¼ Qa þ Qb ;
ð12Þ
_ ¼ ke q_ e ; Q
ð13Þ
Qa q_ a0 ¼ Q0a q_ p ;
ð14Þ
_ b ¼ kp q_ p ; Q
ð15Þ
kQa k 6 Q0a ;
ð16Þ
q_ a0 P 0;
ð17Þ
kQa kq_ a0 ¼ Q0a q_ a0 :
ð18Þ
In Eqs. (1) and (2), p a1 :¼ sin h þ ; 3
p a3 :¼ cos h þ ; 3
ð19Þ pffiffiffi 11 where h can be any real number. If choosing h ¼ 0 we have the stress space Q :¼ ð 3s =2; s11 =2 þ s22 ; t s23 ; s13 ; s12 Þ of IlÕyushin (1963). Throughout this paper, a superscript t indicates the transpose, and a superimposed dot denotes a differentiation with respect to time. In the tensorial representation of elastoplastic model with PragerÕs kinematic hardening rule combined with isotropic hardening rule as shown by Eqs. (3)–(10), e, ee , ep , s, sa and sb are, respectively, the deviatoric tensors of strain, elastic strain, plastic strain, stress, active stress and back stress, all symmetric and traceless, whereas k is a scalar evaluated by Z t pffiffiffi kðtÞ ¼ 2k_ep ðnÞk dn; ð20Þ a2 :¼ sin h;
a4 :¼ cos h;
0
pffiffiffiffiffiffiffiffiffiffiffiffi where k_e k :¼ e_ p e_ p defines the Euclidean norm of e_ p and a dot between two tensors stands for their inner product. From Eqs. (1) and (2) it follows that p
2
2
ksa k ¼ 2kQa k ;
sa e_ ¼ 2Qta q_ :
ð21Þ
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The material functions and plastic multipliers in the above two equivalent representations have the following relations: ke ¼ 2G;
kp ¼ 2k 0 ;
Q0a ¼ s0y ;
2q_ a0 ¼ k_:
ð22Þ ffi pffiffiffiffiffiffiffiffiffi The norm of an n-vector Q is defined as kQk :¼ Qt Q. Depending on the number of nonzero stress components in Eq. (1) (and correspondingly nonzero strain components in Eq. (2)) which we consider for a physical problem, for example, the axial tension-compression problem, the biaxial tension-compressiontorsion problem, etc., the dimension n may be an integer with 1 6 n 6 5, and no matter which case is we use n to denote the physical problem dimension. The bold-faced letters q, qe , qp , Q, Qa and Qb are, respectively, the n-vectors of generalized strain, generalized elastic strain, generalized plastic strain, generalized stress, generalized active stress and generalized back stress, whereas qa0 is a scalar. All q, qe , qp , Qa , Qb , Q and qa0 are functions of one and the same independent variable, which in most cases is taken either as the usual time or as the arc length of the controlled generalized strain path; however, for convenience, the independent variable no matter what it is will be simply called ‘‘time’’ and given the notation t. The generalized elastic modulus ke > 0, generalized kinematic modulus kp , and the generalized yield strength Q0a > 0 are the only three property functions needed in the model, and are functions of the equivalent generalized plastic strain qa0 , which by Eqs. (14), (17) and (18) and Q0a > 0 is found to be Z t Z t kq_ p ðnÞk dn: ð23Þ q_ a0 ðnÞ dn ¼ qa0 ¼ 0
0
The material is further assumed to be weakly stable (Hong and Liu, 1993): 0
ke þ kp þ Q0a > 0;
ke þ kp > 0:
ð24Þ
Here a prime denotes the derivative of a function with respect to its argument, for example, 0 Q0a :¼ dQ0a ðqa0 Þ=dqa0 .
3. Response operators From Eqs. (11)–(15) it follows that _ a þ ke þ kp q_ a Qa ¼ ke q_ ; Q 0 Q0a which, in terms of the integrating factor Z qa
0 k ðpÞ þ k ðpÞ e p a Y ðq0 Þ :¼ exp dp ; Q0a ðpÞ 0
ð25Þ
ð26Þ
can be integrated to Qa ðtÞ ¼
Z t 1 a a a _ Y ðq ðt ÞÞQ ðt Þ þ k ðq ðnÞÞY ðq ðnÞÞ q ðnÞ dn : e 0 a i 0 i 0 Y ðqa0 ðtÞÞ ti
Upon substituting Eq. (27) for Qa into Eq. (14) and integrating the resultant, we obtain Z t Gp ðqa0 ðtÞ; qa0 ðti ÞÞ qp ðtÞ ¼ qp ðti Þ þ Q ðt Þ þ Gp ðqa0 ðtÞ; qa0 ðnÞÞq_ ðnÞ dn; a i ke ðqa0 ðti ÞÞ ti
ð27Þ
ð28Þ
C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
255
where Cðqa0 Þ :¼
Z
qa0
0
1 dp; Y ðpÞQ0a ðpÞ
Gp ðp1 ; p2 Þ :¼ ke ðp2 ÞY ðp2 Þ½Cðp1 Þ Cðp2 Þ:
ð29Þ ð30Þ
Integrating Eq. (13) after replaced its q_ e by q_ q_ p due to Eq. (11), q_ p by the flow rule (14) and Qa by Eq. (27), we obtain Z t Qa ðti Þ þ QðtÞ ¼ Qðti Þ þ Gs ðqa0 ðtÞ; qa0 ðti ÞÞ Gs ðqa0 ðti Þ; qa0 ðti ÞÞ Gs ðqa0 ðtÞ; qa0 ðnÞÞq_ ðnÞ dn; ð31Þ ke ðqa0 ðti ÞÞ ti where Ce ðqa0 Þ
Z
qa0
:¼ 0
ke ðpÞ dp; Y ðpÞQ0a ðpÞ
Gs ðp1 ; p2 Þ :¼ ke ðp2 Þf1 Y ðp2 Þ½Ce ðp1 Þ Ce ðp2 Þg:
ð32Þ ð33Þ
Up to now three integral operators (27), (28) and (31) are established which map respectively generalized strain rate histories into generalized active stress histories, generalized plastic strain histories and generalized stress histories. Here t is the (current) time and ti is an initial time, at which initial conditions Qa ðti Þ, qp ðti Þ, Qðti Þ and qa0 ðti Þ should be prescribed. Obviously, the expressions in Eqs. (27), (28) and (31) that the responses are in terms of the generalized strain rate history is useful only if the history of qa0 is already known. Thus qa0 deserves a more study, as will be pursued in the following two sections.
4. Switch of plastic irreversibility The complementary trios (16)–(18) enable the model to possess a switch of plastic irreversibility, the criteria for which are derived right below. Taking the inner product of Eq. (25) with Qa , we get bQta q_ ¼ Q0a q_ a0
if kQa k ¼ Q0a ;
ð34Þ
where b :¼
ke >0 ke þ kp þ Q0a0
ð35Þ
is a material function. Since b > 0 and Q0a > 0, Eq. (34) assures that if kQa k ¼ Q0a
then
Qta q_ > 0 () q_ a0 > 0:
ð36Þ
Hence, fkQa k ¼ Q0a and Qta q_ > 0g ) q_ a0 > 0:
ð37Þ
On the other hand, if q_ a0 > 0, Eq. (18) assures kQa k ¼ Q0a , which together with Eq. (36) asserts that q_ a0 > 0 ) fkQa k ¼ Q0a and Qta q_ > 0g:
ð38Þ
Therefore, from Eqs. (37) and (38) we conclude that the yield condition kQa k ¼ Q0a and the straining condition Qta q_ > 0 are sufficient and necessary for plastic irreversibility q_ a0 > 0.
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Considering this and the inequality (17), we thus reveal the following criteria of plastic irreversibility: ( t b _ > 0 if kQa k ¼ Q0a and Qta q_ > 0; 0 Q q a q_ 0 ¼ Qa a ð39Þ 0 if kQa k < Q0a or Qta q_ 6 0: In the phase of the switch, q_ a0 > 0, the mechanism of plastic irreversibility is working and the material exhibits elastoplastic behavior, while in the phase of the switch, q_ a0 ¼ 0, the material behavior is reversible and elastic. According to the complementary trios (16)–(18), there are just two phases: q_ a0 > 0 0 a 0 and kQa k ¼ Qa , and corresponds to the q_ 0 ¼ 0 and kQa k 6 Qa . From the switch (39) it is clear that phase whereas to the phase. 5. Measure of plastic irreversibility The switch (39) together with the relation (26) reveals the core importance of qa0 and Y in the whole model; we now go further to investigate their evolutions. Substitution of Eq. (27) into Eq. (34) and rearrangement yield Z t 1 0 a t a YQ q_ ¼ Y ðq0 ðti ÞÞQa ðti Þq_ ðtÞ þ ke ðqa0 ðnÞÞY ðqa0 ðnÞÞq_ t ðtÞq_ ðnÞ dn: ð40Þ b a 0 ti Let Zðqa0 Þ :¼
Z
qa0 0
Y ðpÞQ0a ðpÞ dp; bðpÞ
ð41Þ
such that 0
YQa a Z_ ¼ Z 0 q_ a0 ¼ q_ : b 0
ð42Þ
From the inequality (17) and Q0a > 0, b > 0, and Y > 0 in view of the definition (26), we have Z_ P 0;
Z0 ¼
YQ0a > 0; b
ð43Þ
which ensures the invertibility of the material function Zðqa0 Þ. Hence, qa0 ¼ qa0 ðZÞ
ð44Þ
is available, and all the material functions that originally expressed in terms of qa0 can be re-expressed in terms of Z. However, for saving notations we use the same symbols to denote such new functions, for example, Y ðZÞ :¼ Y ðqa0 ðZÞÞ:
ð45Þ
Substituting Eqs. (42) and (44) into Eq. (40), we obtain Z t Z_ ðtÞ ¼ Y ðZðti ÞÞQta ðti Þq_ ðtÞ þ ke ðZðnÞÞY ðZðnÞÞq_ t ðtÞq_ ðnÞ dn:
ð46Þ
ti
Then, integration gives ZðtÞ ¼ Zðti Þ þ Y ðZðti ÞÞQta ðti Þ½qðtÞ qðti Þ þ
Z
t
ke ðZðnÞÞY ðZðnÞÞ½qðtÞ qðnÞt q_ ðnÞ dn;
ti
which is a nonlinear Volterra integral equation for Z.
ð47Þ
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257
6. Homeomorphisms of measures of plastic irreversibility From Eqs. (26), (42), (43)1 and (24)2 it follows that bðke þ kp Þ _ Z P 0; Y_ ¼ 2 ðQ0a Þ
ð48Þ
and the dissipation power is K_ :¼ Qta q_ p ¼ Q0a q_ a0 P 0:
ð49Þ
In view of the switch (39), inequalities (17) and (43)1 , and Eqs. (42), (48) and (49), it is important to note that qa0 , Y , Z and K are intimately related in the sense that there exists strictly monotonic increasing relation between any pair of them as shown in Table 1. Therefore, any one of those irreversibility parameters can be chosen to play the role in the switch (39), the role of an indicator of irreversible change of material properties, the role of so-called material age, intrinsic time, internal time, the arrow of time, etc., in this time-independent model. Furthermore, they serve as the measures of plastic irreversibility and are crucially important in the evolutions of material properties and parameters and also in the determination of responses as shown in Eqs. (27), (28) and (31). Once Z is obtained, the other parameters are readily calculable through the integrals of the material homeomorphic functions listed in Table 1. Hence, Eq. (47) or Eq. (46) deserves a further study.
7. Material functions in terms of Z By Eqs. (41) and (44) the irreversibility parameter Z and the equivalent generalized plastic strain qa0 are closely related, being bijective, invertible, and strictly monotonic. This fact of equivalence can be utilized to accelerate the calculation of the material responses. For the generalized strain-controlled processes we may thereby change the dependence of the defined material functions on qa0 to directly on Z, and thus all the material functions with the arguments qa0 turn to the material functions simply with the arguments Z through Eq. (44). Even the process listed in the above is available to reveal the function dependence of Q0a ðZÞ, kp ðZÞ and so on. But they are not so straightforward. In this section the relationships of the material functions are further studied from a different point-of-view. Table 1 Relationships between irreversibility parameters q_ a Y_ 0
q_ a0
1
Q0a ðke þ kp ÞY
Y_
ðke þ kp ÞY Q0a
1
Z_ K_
Q0a Y b Q0a
X_ a0
ke Y b
X_ c0
ke Y ½1 2q0 Xc0 2 b½1 2q0 Q0a ð0Þ
ðQ0a Þ2 bðke þ kp Þ
Z_
K_
X_ a0
X_ c0
b Q0a Y
1 Q0a
b ke Y
ke Y ½1 2q0 Xc0 2
bðke þ kp Þ
ðke þ kp ÞY ðQ0a Þ2
bðke þ kp Þ ke Q0a
bðke þ kp Þ½1 2q0 Q0a ð0Þ
ðQ0a Þ2 Y b
Q0a ke
Q0a ½1 2q0 Q0a ð0Þ bQ0a ½1 2q0 Q0a ð0Þ
1
ðQ0a Þ2 ðke þ kp ÞY
b Y
1
bQ0a ke Y
ke Q0a
ke Q0a
ke Y bQ0a
1
ke ½1 2q0 Xc0 2 Q0a ½1 2q0 Q0a ð0Þ
ke Y ½1 2q0 Xc0 2 bQ0a ½1 2q0 Q0a ð0Þ
½1 2q0 Xc0 2 1 2q0 Q0a ð0Þ
bðke þ kp Þ 2q0 Xc0 2 kp Þ½1 2q0 Q0a ð0Þ
ke Q0a ½1 bðke þ
b½1 2q0 Q0a ð0Þ
ke Q0a ½1 2q0 Xc0 2 ke ½1 2q0 Xc0 2 ke Y ½1 2q0 Xc0 2 1 2q0 Q0a ð0Þ ½1 2q0 Xc0 2 1
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C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
In order to give the dependence of the material functions on Z directly, we combine Eqs. (45) and (26) to get Z qa
0 k ðpÞ þ k ðpÞ e p exp dp ¼ Y ðZÞ: ð50Þ Q0a ðpÞ 0 Derivating both sides with respect to qa0 , applying the chain rule on the right-hand side and then using Eq. (43)2 , we obtain bðqa0 Þ½ke ðqa0 Þ þ kp ðqa0 Þ ðQ0a ðqa0 ÞÞ
¼ Y 0 ðZÞ:
2
ð51Þ
Similarly, the derivative of Eq. (32) with respect to qa0 and the use of Eq. (43)2 give bðqa0 Þke ðqa0 Þ ðQ0a ðqa0 ÞÞ
2
¼ Y 2 ðZÞCe0 ðZÞ:
ð52Þ
By dividing the above two equations we obtain ke ðZÞY 0 ðZÞ ; Y 2 ðZÞCe0 ðZÞ
ð53Þ
Y 0 ðZÞ 1 : Y 2 ðZÞCe0 ðZÞ
ð54Þ
ke ðqa0 Þ þ kp ðqa0 Þ ¼ or in terms of Z, kp ðZÞ ¼ ke ðZÞ
The replacement of b in Eq. (52) by the one in Eq. (35) and the substitution of Eq. (53) into the resultant yield
ke2 ðZÞ ke ðZÞY 0 ðZÞ 0 a 2 00 a ¼ ðQa ðq0 ÞÞ Qa ðq0 Þ þ 2 : ð55Þ Y 2 ðZÞCe0 ðZÞ Y ðZÞCe0 ðZÞ 0
By the chain rule and the use of Eq. (43)2 the term Q0a ðqa0 Þ in the above equation can be written as 0
0
Q0a ðqa0 Þ ¼ Q0a ðZÞ
Y ðZÞQ0a ðZÞ : bðZÞ
ð56Þ
Upon using Eqs. (53) and (35) it can be arranged to 0
0 Q0a ðqa0 Þ
¼
0
ðZÞY ðZÞ Q0a ðZÞQ0a ðZÞ kYeðZÞC 0 ðZÞ e
ke ðZÞ Y ðZÞQ0a ðZÞQ0a0 ðZÞ
;
which is then substituted into Eq. (55), giving 2 3 ke ðZÞY 0 ðZÞ 0 00 0 Q ðZÞQ ðZÞ 0 ðZÞ ke2 ðZÞ k ðZÞY ðZÞ a a Y ðZÞC e 2 e 5: ¼ ðQ0a ðZÞÞ 4 þ Y 2 ðZÞCe0 ðZÞ ke ðZÞ Y ðZÞQ0a ðZÞQ0a0 ðZÞ Y 2 ðZÞCe0 ðZÞ
ð57Þ
ð58Þ
With some algebraic manipulations the following differential equation relating Y ðZÞ and ðQ0a ðZÞÞ2 can be obtained, d 2Y 0 ðZÞ 0 ke ðZÞ 2 2 ðQ0a ðZÞÞ þ ðQa ðZÞÞ ¼ : dZ Y ðZÞ Y ðZÞ
ð59Þ
This equation provides an interesting relation among the three material functions ke ðZÞ, Q0a ðZÞ and Y ðZÞ.
C.-S. Liu / International Journal of Solids and Structures 40 (2003) 251–269
259
2
If Y ðZÞ and ke ðZÞ are given then the solution of ðQ0a ðZÞÞ obtained from Eq. (59) is Z Z 2 2 ðQ0 ð0ÞÞ 2 ðQ0a ðZÞÞ ¼ 2 ; ke ðpÞY ðpÞ dp þ a2 Y ðZÞ 0 Y ðZÞ and, if Q0a ðZÞ and ke ðZÞ are given then the solution of Y ðZÞ obtained from Eq. (59) is Z Z 1 ke ðpÞ Q0a ð0Þ dp þ : Y ðZÞ ¼ 0 Qa ðZÞ 0 Q0a ðpÞ Q0a ðZÞ Ce ðZÞ is calculated from Eq. (54) by Z Z ke ðpÞY 0 ðpÞ dp; Ce ðZÞ ¼ 2 0 ½ke ðpÞ þ kp ðpÞY ðpÞ
ð60Þ
ð61Þ
ð62Þ
and kp ðZÞ is evaluated by Eq. (54). Similarly, Z Z 0 Ce ðpÞ dp CðZÞ :¼ ke ðpÞ 0
ð63Þ
is obtained from Eq. (29) by changing the integration variable by means of Eqs. (42) and (52). 2 Eq. (60) provides a functional relation between ðYQ0a Þ and ke Y , and Eq. (61) a functional relation between YQ0a and ke =Q0a . The above two equations are consistent. Interestingly, the parameter YQ0a has already appeared in several places, for example, Eqs. (29), (32), (41)–(43), etc. In Section 9 we will reveal the importance of the parameter YQ0a for further understanding of the mixed-hardening plasticity model.
8. Operators in terms of Z Once Z is obtained, the material functions Q0a ðZÞ and kp ðZÞ are specified, and Y ðZÞ and Ce ðZÞ are evaluated via Eqs. (61) and (62) (or Y ðZÞ and Ce ðZÞ are specified, and Q0a ðZÞ and kp ðZÞ are evaluated via Eqs. (60) and (54)), Qa ðtÞ and QðtÞ can be calculated respectively by
Z t 1 Y ðZðti ÞÞQa ðti Þ þ Qa ðtÞ ¼ ke ðZðnÞÞY ðZðnÞÞq_ ðnÞ dn ; ð64Þ Y ðZðtÞÞ ti QðtÞ ¼ Qðti Þ þ ½Gs ðZðtÞ; Zðti ÞÞ Gs ðZðti Þ; Zðti ÞÞ
Qa ðti Þ þ ke ðZðti ÞÞ
Z
t
Gs ðZðtÞ; ZðnÞÞq_ ðnÞ dn;
ð65Þ
ti
where Gs ðZ1 ; Z2 Þ :¼ ke ðZ2 Þf1 Y ðZ2 Þ½Ce ðZ1 Þ Ce ðZ2 Þg: Similarly, the generalized plastic strain is calculated by Z t Gp ðZðtÞ; Zðti ÞÞ qp ðtÞ ¼ qp ðti Þ þ Qa ðti Þ þ Gp ðZðtÞ; ZðnÞÞq_ ðnÞ dn; ke ðZðti ÞÞ ti
ð66Þ
ð67Þ
where Gp ðZ1 ; Z2 Þ :¼ ke ðZ2 ÞY ðZ2 Þ½CðZ1 Þ CðZ2 Þ:
ð68Þ
A numerical procedure based on the discretizations of Eqs. (47), (64), (65) and (67) can be developed to calculate the responses of mixed-hardening elastoplastic model under general loading conditions. However, this important computational issue is reported by Liu (submitted for publication) in other place.
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9. Augmented differential equations system From Eqs. (46) and (64) we have
1
Z_ ¼ Y ðZÞQta q_ ;
ð69Þ
and from Eqs. (25), (26), (44) and (45) we have d ½Y ðZÞQa ¼ ke ðZÞY ðZÞq_ ; dt
ð70Þ
which can be rearranged to d ke ðZÞ ½Y ðZÞQa ¼ Y ðZÞQ0a ðZÞ 0 q_ : dt Qa ðZÞ
ð71Þ
On the other hand, multiplying Eq. (61) by Q0a on both sides, taking the time derivative and then using Eq. (69) we obtain d ke ðZÞ t ½Y ðZÞQ0a ðZÞ ¼ Y ðZÞ 0 Q q_ : dt Qa ðZÞ a Let us introduce s
Xa Y Qa Xa ¼ :¼ Xa0 YQ0a
ð72Þ
ð73Þ
as the augmented (n þ 1)-dimensional state vector, 2 where Xa0 is another irreversible parameter, and Table 1 shows its relations with the other irreversible parameters. Correspondingly, Xsa :¼ Y Qa is the spatial part of Xa in the Minkowski spacetime, and we may call it the augmented generalized active stress. Moreover, from Eqs. (39), (72) and (73) it follows that ( 0 ke Xa Qt q_ > 0 if kQa k ¼ Q0a and Qta q_ > 0; X_ a0 ¼ ðQ0a Þ2 a ð74Þ 0 if kQa k < Q0a or Qta q_ 6 0: Thus, Eqs. (71) and (72) can be combined together as following equations system: X_ a ¼ AXa ;
ð75Þ
where A :¼
ke 0nn Q0a q_ t
q_ : 0
Define the proper time s as follows: Z t ke ke ðnÞ dn; ds ¼ 0 dt; s ¼ si þ 0 ðnÞ Q Qa ti a
ð76Þ
ð77Þ
1 Because Y ðZÞQa is the augmented generalized active stress, we may correspondingly call Z_ the augmented generalized active power. 2 From Eqs. (73) and (61) we have X_ a0 ¼ ke Z_ =Q0a > 0 in the on phase, and thus Xa0 is a time-like parameter which can be viewed as the temporal component of Xa in the Minkowski spacetime.
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where ti is an initial time and si is the corresponding initial proper time. Since the term ke =Q0a in Eq. (76) is positive, the proper time s is a monotonic function of the external time t. The on-off switching criteria in Eq. (74) can thus be written as dXa0 ¼ ds
(
Xa0 Q0a
Qta q_ > 0 if kQa k ¼ Q0a and Qta q_ > 0; 0 if kQa k < Q0a or Qta q_ 6 0:
ð78Þ
In terms of the new time scale s, Eq. (75) becomes d Xa ¼ BXa ; ds
ð79Þ
where
0 B :¼ nn q_ t
q_ : 0
ð80Þ
It can be seen that in this augmented space Xa ¼ ðY Qta ; YQ0a Þt , the governing equations become linear with respect to s and are more tractable than the original nonlinear equations. From Eqs. (79) and (80) a deeper understanding of the underlying structure of the model may be achieved as to be done in Sections 10–12.
10. The Minkowski spacetime In Section 9 we found that even the constitutive equations are nonlinear in the n-dimensional state space of generalized active stresses Qa , but they can be transformed to linear differential equations in the (n þ 1)dimensional augmented state space of Xa through a time scaling. In the augmented space not only the nonlinearity of the model is unfolded, but also an intrinsic spacetime structure of the Minkowskian type will be brought out. Now we translate the mixed-hardening elastoplastic model postulated in Section 1 in the state space of Qa to one in the augmented state space of Xa . Accordingly the first row in Eq. (79), and Eqs. (18), (16) and (17) become
In 0n1 dXa 0nn q_ ¼ ð81Þ X; 01n Xta gXa ds 01n 0 a Xta gXa 6 0;
ð82Þ
dXa0 P 0; ds
ð83Þ
in terms of the Minkowski metric (in the space-like convention)
In 0n1 g :¼ ; 01n 1
ð84Þ
where In is the identity matrix of order n. The vector space of augmented states Xa endowed with the Minkowski metric tensor g is referred to as Minkowski spacetime, and designated as Mnþ1 . Regarding Eqs. (16) and (82), we further distinguish two correspondences: kQa k ¼ Q0a () Xta gXa ¼ 0;
ð85Þ
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kQa k < Q0a () Xta gXa < 0:
ð86Þ
That is, a generalized active stress state Qa on (resp. within) the yield hypersphere kQa k ¼ Q0a in the generalized active stress space of (Q1a ; . . . ; Qna ) corresponds to an augmented state Xa on the cone fXa jXta gXa ¼ 0g of Minkowski spacetime (resp. in the interior fXa jXta gXa < 0g of the cone). The exterior fXa jXta gXa > 0g of the cone is uninhabitable since kQa k > Q0a is forbidden. Even though it admits an infinite number of Riemannian metrics, the yield hypersphere Sn1 of Qa does not admit a Minkowskian metric, nor does the space of (Qta ; Y ). It is the cone of Xa which admits the Minkowski metric. From Eqs. (34), (73) and (84) it follows that t
kQa k ¼ Q0a ) Xta gðq_ t ; q_ a0 =bÞ ¼ 0:
ð87Þ
Moreover, by Eqs. (73), (78)1 , (79) and (84) we can prove that fkQa k ¼ Q0a and Qta q_ > 0g ) Xta g
dXa ¼ 0: ds
ð88Þ
If the model is in the on phase (i.e., not only kQa k ¼ Q0a but also Qta q_ > 0), then from Eqs. (85), (87) and (88) we assert that for an Xa -path on the cone, the augmented state Xa is M-orthogonal to itself, its tangent vector dXa =ds and also its dual ðq_ t ; q_ a0 =bÞ. The so called M-orthogonality is an orthogonality of two (n þ 1)vectors with respect to metric (84) in Minkowski spacetime Mnþ1 (see, e.g., Das, 1993 and Naber, 1992). On the other hand, Xa0 is frozen in the off phase as indicated by Eq. (78)2 and the augmented state Xa stays in the closed n-disc Dn (i.e., closed n-ball Bn ) on the hyperplane Xa0 ¼ constant in the space of (Xa1 ; . . . ; Xan ; Xa0 ), the hyperplane being identified to be Euclidean n-space En , which is endowed with the Euclidean metric In . In summary, the augmented state Xa either evolves on the cone when in the on phase or stays in the discs of simultaneity, which are stacked up one by one in the interior of the cone and are glued to the cone, when in the off phase.
11. Space-like paths in the Minkowski spacetime The criteria in Eq. (39) ensure that _ a Q0 Q00 q_ a ¼ 0 q_ a0 ½Qta Q a a 0
ð89Þ
no matter whether in the on or in the off phase. Substituting Eqs. (11)–(15) into the above equation and using Eq. (23), we obtain bq_ t q_ p ¼ ðq_ p Þt q_ p P 0:
ð90Þ
Materials which satisfy such an inequality q_ t q_ p P 0 are said to be kinematically stable, e.g., Lubliner (1984). Using Eq. (23) and the above equation we obtain pffiffiffiffiffiffiffiffiffiffiffi ð91Þ q_ a0 ¼ bq_ t q_ p : Squaring of the above equation and using Eqs. (17) and (14), we get 2
ðq_ a0 Þ ¼ q_ a0
bQta q_ ; Q0a
ð92Þ
by which we have 2
ðq_ a0 Þ 6
bq_ a0 kQa kkq_ k; Q0a
ð93Þ
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and using Eq. (18) the bound of q_ a0 can be derived, q_ a0 6 bkq_ k
ð94Þ
no matter in the on or off phase. This inequality tells us that the maximum value of the specific dissipation power K_ ¼ Q0a q_ a0 an admissible path in the state space may discharge is bQ0a kq_ k. (On the other hand, postulation (17) tells us that the minimum value of the specific dissipation power an admissible path may discharge is zero.) The solution of Eq. (79) may be viewed as a path in Mnþ1 , which traces along the null cone Xta gXa ¼ 0 given by Eq. (85). From which the derivative gives Xta g dXa ¼ 0:
ð95Þ
Thus Xa is M-orthogonal to dXa . Furthermore, upon using this equation and 2 !2 3 n n n n X X X X 1 4 2 2 2 2 ðdXai Þ ðdXa0 Þ ¼ ðXai Þ ðdXai Þ Xai dXai 5: 0 Þ2 ðX i¼1 i¼1 i¼1 i¼1 a
Xta gXa
¼ 0 we have ð96Þ
So that by utilizing the Schwartz inequality we obtain 2
t
ðdXa Þ :¼ ðdXa Þ g dXa P 0:
ð97Þ t
Recalling that a path such that ðdXa Þ g dXa > 0 (resp: ¼ 0; < 0) is called a space-like (resp. null, timelike) path in Mnþ1 , we thereby conclude that the path Xa ðs0 Þ, si < s0 6 s, in the augmented state space is a space-like or null path in the Minkowski spacetime Mnþ1 no matter in the on or in the off phase. Indeed, Eq. (97) conveys a message that the nature of mixed-hardening elastoplasticity rejects time-like paths, so that the time-like metric convention has to be rejected to avoid an unreasonable negative squared length. This is the reason why we have adopted the space-like convention (84) for mixed-hardening elastoplasticity.
12. The proper orthochronous Lorentz group In this section we concentrate on the on phase to bring out internal symmetry inherent in the model in the on phase. Denote by Ion an open, maximal, continuous proper time interval during which the mechanism of plasticity is on exclusively. The solution of the augmented state equation (79) can be expressed as in the following augmented state transition formula: Xa ðsÞ ¼ ½GðsÞG1 ðs1 ÞXa ðs1 Þ;
8 s; s1 2 Ion ;
ð98Þ
in which GðsÞ, called the fundamental matrix of Eq. (79), is a square matrix of order n þ 1 satisfying d GðsÞ ¼ BðsÞGðsÞ; ds Gð0Þ ¼ Inþ1 :
ð99Þ ð100Þ
On the other hand, from Eqs. (80) and (84) it is easy to verify that the control matrix B in the on phase satisfies Bt g þ gB ¼ 0: By Eqs. (101) and (99) we find d t ½G ðsÞgGðsÞ ¼ 0: ds
ð101Þ
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From Eq. (100) we have Gt ðsÞgGðsÞ ¼ Itnþ1 gInþ1 ¼ g at s ¼ 0, and thus prove that Gt ðsÞgGðsÞ ¼ g
ð102Þ
for all s 2 Ion . Take determinants of both sides of the above equation, getting 2
ðdet GÞ ¼ 1;
ð103Þ
so that G is invertible. The 00 entry of the matrix equation (102) is ðG00 Þ2 P 1:
Pn
i 2 i¼1 ðG0 Þ
2
ðG00 Þ ¼ 1, from which ð104Þ
Here Gij ; i; j ¼ 1; . . . ; n; 0; is the mixed ij-entry of the matrix G. Since det G ¼ 1 or G00 6 1 would violate Eq. (100), it turns out that det G ¼ 1;
ð105Þ
G00 P 1:
ð106Þ
In summary, G has the three characteristic properties explicitly expressed by Eqs. (102), (105) and (106). Recall that the complete homogeneous Lorentz group Oðn; 1Þ is the group of all invertible linear transformations in Minkowski spacetime which leave the Minkowski metric invariant, and that the proper orthochronous Lorentz group SOo ðn; 1Þ is a subgroup of Oðn; 1Þ in which the transformations are proper (i.e., orientation preserving, namely the determinants of the transformations being þ1) and orthochronous (i.e., time-orientation preserving, namely the 00 entry of the matrix representations of the transformations being positive) (see, e.g., Cornwell, 1984). Hence, in view of the three characteristic properties we conclude that the fundamental matrix G belongs to the proper orthochronous Lorentz group SOo ðn; 1Þ. So the matrix function GðsÞ of proper time s 2 Ion may be viewed as a connected path of the Lorentz group. Furthermore, by Eq. (101), B is an element of the real Lie algebra soðn; 1Þ of the Lorentz group SOo ðn; 1Þ. From Eq. (78)1 , dXa0 =ds > 0 strictly when the mechanism of plasticity is on; hence, 3 Xa0 ðsÞ > Xa0 ðs1 Þ > Xa0 ðs0 Þ ¼ Q0a ;
8s > s1 > s0 ;
s; s1 2 Ion ;
ð107Þ
which means that in the sense of irreversibility there exists future-pointing proper time-orientation from the augmented states Xa ðs1 Þ to Xa ðsÞ. Moreover, such time-orientation is a causal one, because the augmented state transition formula (98) and inequality (107) establish a causality relation between the two augmented states Xa ðs1 Þ and Xa ðsÞ in the sense that the preceding augmented state Xa ðs1 Þ influences the following augmented state Xa ðsÞ according to formula (98). Accordingly, the augmented state Xa ðs1 Þ chronologically and causally precedes the augmented state Xa ðsÞ. This is indeed a common property for all models with inherent symmetry of the proper orthochronous Lorentz group. By this symmetry a core connection among irreversibility, the time arrow of evolution, and causality has thus been established for plasticity in the on phase.
13. The Poincare group The proper orthochronous Lorentz group SOo ðn; 1Þ constructed in the above is acting on the space Xa ; and what is the effect of the kinematic hardening on the group structure? In order to reply this question we 3 From Eqs. (17), (26) and (73) it follows that Xa0 ðsÞ P Xa0 ðs0 Þ P Xa0 ðsi Þ P Xa0 ðs0 Þ ¼ Q0a for all s P s0 P si P s0 , applicable to both the on and off phases. Recall that s0 is the zero-value proper time at which all relevant values including qa0 ðs0 Þ ¼ 0.
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return to the derivation of Eq. (31) as follows. First, from Eqs. (15), (14) and (27), the generalized back stress is integrated as follows: Z t Gb ðqa0 ðtÞ; qa0 ðti ÞÞ Qa ðti Þ þ Qb ðtÞ ¼ Qb ðti Þ þ Gb ðqa0 ðtÞ; qa0 ðnÞÞq_ ðnÞ dn; ð108Þ ke ðqa0 ðti ÞÞ ti where Z
qa0
kp ðpÞ dp; Y ðpÞQ0a ðpÞ
ð109Þ
Gb ðp1 ; p2 Þ :¼ ke ðp2 ÞY ðp2 Þ Cp ðp1 Þ Cp ðp2 Þ :
ð110Þ
Cp ðqa0 Þ :¼
0
Both Cp and Gb are material functions. In terms of Ga ðp1 ; p2 Þ :¼
ke ðp2 ÞY ðp2 Þ ; Y ðp1 Þ
ð111Þ
it can be verified from Eqs. (26), (32), (33), (110) and (109) that Gs ðp1 ; p2 Þ :¼ ke ðp2 Þf1 Y ðp2 Þ½Ce ðp1 Þ Ce ðp2 Þg ¼ Ga ðp1 ; p2 Þ þ Gb ðp1 ; p2 Þ:
ð112Þ
Combining Eqs. (27) and (108) and noting Eqs. (12) and (112) yield Eq. (31) again. Equation (108) is an integral representation expressing the generalized back stress in terms of the generalized strain rate history. Now, introduce the space X with the following vector decomposition:
Y Qa Qb X ¼ Xa þ Xb :¼ : ð113Þ þ YQ0a 0 The group acting on the space X is the semi-direct product of the translation Tnþ1 and the proper orthochronous Lorentz group SOo ðn; 1Þ; usually such group is named the Poincare group ISOo ðn; 1Þ, or inhomogeneous Lorentz group (see, e.g., Kim and Noz, 1986).
14. Conformal spacetime From Eqs. (11)–(13), (15) and (91) we have 2
2
0
2
2
ðdQa Þ :¼ kdQa k ¼ ke2 kdqk ðke þ kp Þ½ke þ kp þ 2Q0a ðdqa0 Þ ;
ð114Þ
which, with the aid of Eqs. (71)–(73), (39), (84) and (97), leads to 0
ðdXa Þ2 ¼ Y 2 ½ðdQa Þ2 ðQ0a Þ2 ðdqa0 Þ2
ð115Þ 00
no matter in the on or in the off phase. Especially, in the on phase we have Qa dQa ¼ Q0a Qa dqa0 by the 0 plastic consistency condition, and for the nonperfect case, i.e., Q0a 6¼ 0, we have " # " # 2 ðQ dQ Þ 1 0 2 2 2 2 2 a a ðdQa Þ ðdqa0 Þ : ðdXa Þ ¼ Y 2 kdQa k ¼ Y 2 ðQ0a Þ ð116Þ 2 2 ðQ0a Þ ðQ0a0 Þ 2
0
2
2
The metric line element ðdQa Þ =ðQ0a Þ ðdqa0 Þ defined in the space (Qta ; qa0 ) indicates that the spatial sec0 tions expand (or contract) uniformly as described by the scalar function Q0a ðqa0 Þ. This form of the metric line element is manifestly conformal to the Minkowski space Xa with a conformal scalar factor Y ðqa0 Þ. The space (Qta ; qa0 ) is known as a Robertson–Walker spacetime (see, e.g., Hawking and Ellis, 1973 and Birrell and Davies, 1982).
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In this occasion we would like to point out that the metric considered by De Saxce and Hung (1985), who attempted to study plasticity models from a differential geometry view, is not a suitable one. Really, by 2 identifying ðQ0a Þ ¼ 2 r2 =3, dK ¼ Q0a dqa0 ¼ dj0 and dqpa ¼ dja with the notations used by De Saxce and Hung, we readily obtain 2
dr2 ¼ ðdj0 Þ
2 r2 dab dja djb 3
from the flow rule (14). See also Eq. (5.2) proposed by De Saxce and Hung, loc.cit. The above metric vanishes no matter whether in the on or in the off phase. In this sense such metric is not a suitable measure for its full degeneracy. More precisely, it is not a metric and also not a hyperbolic metric. So, De Saxce and Hung asserting it as a hyperbolic metric on the base manifold is incorrect. Indeed, dr2 ¼ 0 is at most an identity derived from the associated flow rule. t Eq. (75) is the governing equation of Xa ¼ ðY Qta ; YQ0a Þ ; and from Eqs. (71), (72), (26), (77) and (39) the governing equation of ðQa ; Q0a Þ is read as
d Qa ke 0nn q_ Qa bðke þ kp Þ t Qa _ Qa q 0 : ¼ 0 ð117Þ 2 Qa 0 Q0a dt Q0a Qa q_ t ðQ0a Þ t
However, ðQta ; Q0a Þ is not a suitable spacetime structure, since dQ0a may be negative in the softening range of modeled material. In order to obtain a suitable conformal spacetime structure, we introduce a scaling factor function qðqa0 Þ and consider the following augmented vector: s
Xc qQa Xc ¼ :¼ ; ð118Þ Xc0 qQ0a in which we need dðqQ0a Þ > 0 in the on phase, and simultaneously the cone is preserved, i.e., ðXc Þt gXc ¼ 0:
ð119Þ
Taking the time derivative of Eq. (118) and using (117) we obtain
d qQa ke 0nn q_ qQa bðke þ kp Þ q_ qQa qQa t _ q qQ ¼ þ : a qQ0a 0 qQ0a dt qQ0a q qQ0a Q0a q_ t qðQ0 Þ2
ð120Þ
a
If q satisfies
ke bðke þ kp Þ q_ t _; ¼ 0 2q 0 qQa q qke Q0a q Qa
ð121Þ
where q0 is a constant whose range to be determined below, then in the space Xc the governing equations are given by
d qQa ke 0nn q_ qQa 2ke q0 qQa t _ q qQ ¼ : ð122Þ a qQ0a 0 qQ0a dt qQ0a Q0a q_ t Q0a By using the proper time
d qQa 0 ¼ nn q_ t ds qQ0a
s defined in Eq. (77) the above equations are further refined to
qQa q_ qQa t _ q qQ 2q : 0 a qQ0a 0 qQ0a
ð123Þ
The above differential equations system is one of the conformal type, e.g., Anderson et al. (1982), with the (n þ 1)-dimensional conformal parameters (q0 q_ ; 0). The group generated from the above differential equations system is a special type of conformal group.
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Eq. (121), after substituting q_ ¼ q0 q_ a0 and Eq. (39)1 for Qta q_ ¼ Q0a q_ a0 =b and cancelling q_ a0 Õs on both sides, can be changed to the following Riccati differential equation: q0 ¼
ke þ kp 2ke q0 2 q: q Q0a b
ð124Þ
Using Eq. (26) the solution of q is found to be qðqa0 Þ ¼
1 þ 2q0
Y ðqa0 Þ R qa0 ke ðpÞY ðpÞ 0
bðpÞ
ð125Þ
: dp
On the other hand, from Eqs. (61) and (42) we have dðYQ0a Þ ke _ ke Y a ¼ 0Z ¼ q_ > 0 dt b 0 Qa
ð126Þ
in the on phase, such that the integral in Eq. (125) can be obtained explicitly, qðqa0 Þ ¼
Y ðqa0 Þ : 1 2q0 Q0a ð0Þ þ 2q0 Y ðqa0 ÞQ0a ðqa0 Þ
ð127Þ
When q0 ¼ 0 we have q ¼ Y , and the differential equations system (79) is recovered from Eq. (123). We also note that q0 can not be a negative constant value; otherwise, the above q will blow-up when Y increases to a certain positive value. From Eq. (127) the relation of Xa and Xc through Eqs. (73) and (118) is available, Xc ¼
1 Xa : 1 2q0 Q0a ð0Þ þ 2q0 Xa0
ð128Þ
Furthermore, the metrics in both the spaces Xc and Xa can be proved to be related conformally as follows: 2
ðdXc Þ ¼
½1 2q0 Xc0 ½1
2
1
2
2 2q0 Q0a ð0Þ
ðdXa Þ ¼
½1
2
2q0 Q0a ð0Þ
2
þ 2q0 Xa0
ðdXa Þ ;
ð129Þ
where 2
t
ðdXc Þ :¼ ðdXc Þ g dXc :
ð130Þ
Finally, from Eq. (128) it is easy to prove that X_ c0 ¼
½1 2q0 Q0a ð0Þ ½1 2q0 Q0a ð0Þ þ
2q0 Xa0 2
2
½1 2q0 Xc0 _ 0 X : X_ a0 ¼ 1 2q0 Q0a ð0Þ a
ð131Þ
For the purpose of Xc0 > 0 to be an irreversible parameter as well, i.e., X_ c0 > 0, by Eqs. (126) and (73) we require 1 2q0 Q0a ð0Þ > 0, that is, 0 6 q0 <
1 : 2Q0a ð0Þ
ð132Þ
Thus, under this condition we have introduced another irreversible parameter Xc0 as the temporal component of the conformal spacetime, the relations of which to the other irreversible parameters are listed in Table 1. It includes two dimensionless irreversible parameters qa0 and Y , and also four stress dimension irreversible parameters Z, K, Xa0 and Xc0 . The hyperbolic geometric models that we studied for the mixed-hardening elastoplasticity are governed by the conformally isormorphic spacetimes. They are all differentiable manifold endowed with a
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pseudo-Riemann metric, which assigns at each spacetime point Xc an indefinite symmetric inner product on the tangent space, which varying differentiably with Xc .
15. Conclusions In this paper we have established two types formulations for mixed-hardening elastoplastic model of the generalized stress and generalized strain: the flow model characterized by Eqs. (11)–(18) and the two-phase linear differential system characterized by Eqs. (79) and (80). Even though the governing equations of the generalized active stress are nonlinear in the n-dimensional space Qa we have found that it can be transformed to linear differential equations in the augmented (n þ 1)-dimensional space Xa . In this space not only the nonlinearity of the model can be unfolded, but also an intrinsic spacetime structure of the Minkowskian type can be revealed, merely replacing the inequality in space Qa to the inequality in the augmented space Xa . We also pointed out that the state matrix B is an element of the Lie algebra of the proper orthochronous Lorentz group, hence the state transition matrix generated from the linear differential equations was proved to be a type of the proper orthochronous Lorentz group. In the frame of the Minkowski spacetime we have further proved that the action of the kinematic rule in the mixed-hardening model causes a translation of the state X, which amounts to extend the proper orthochronous Lorentz group to the proper orthochronous Poincare group. We have introduced a conformal factor q in Eq. (127) and thus a conformal spacetime Xc is derived. The analytic models of hyperbolic geometry that we studied for the mixed-hardening elastoplasticity are governed by the conformally isormorphic spacetimes. They are all differentiable manifold endowed with a pseudo-Riemann metric, which assigns at each spacetime point an indefinite symmetric inner product on the tangent space, which varying differentiably. Mathematically speaking, the pseudo-Riemannian manifold is a suitable underlying spacetime model for the mixed-hardening elastoplasticity. Even we dealt only with mixed-hardening effect on the elastoplastic model without considering a more inclusive Armstrong-Frederick kinematic hardening rule, 4 and/or large deformation, etc.; however, we may further consider more sophisticated group actions on pseudo-Riemannian manifolds and thus makes an explicit use of the powerful group-theoretic method to study plasticity from a global view. In this regard, the present paper may open the way to plasticity research in a new direction.
Acknowledgement The financial support provided by the National Science Council under Grant NSC 90-2212-E-019-007 is gratefully acknowledged.
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4 _ b ¼ kp q_ p kb q_ a Qb with kb > 0 to account Armstrong and Frederick (1966) have proposed a nonlinear kinematic hardening rule Q 0 of the saturation of Qb under proportional loading conditions.
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